Method for predicting resist deformation

ABSTRACT

A method for determining a deformation of a resist in a patterning process. The method involves obtaining a resist deformation model of a resist having a pattern, the resist deformation model configured to simulate a fluid flow of the resist due to capillary forces acting on a contour of at least one feature of the pattern; and determining, via the resist deformation model, a deformation of a resist pattern to be developed based on an input pattern to the resist deformation model.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority of U.S. application 62/786,637 whichwas filed on Dec. 31, 2018 and which is incorporated herein in itsentirety by reference.

TECHNICAL FIELD

The description herein relates to processes involving pattern formationon a substrate, and more particularly to a method of determining resistdeformation of a patterned layer on the substrate.

BACKGROUND

A lithographic apparatus can be used, for example, in the manufacture ofintegrated circuits (ICs) or other devices. In such a case, a patterningdevice (e.g., a mask) may contain or provide a pattern corresponding toan individual layer of the device (“design layout”), and this patterncan be transferred onto a target portion (e.g. comprising one or moredies) on a substrate (e.g., silicon wafer) that has been coated with alayer of radiation-sensitive material (“resist”), by methods such asirradiating the target portion through the pattern on the patterningdevice. In general, a single substrate contains a plurality of adjacenttarget portions to which the pattern is transferred successively by thelithographic apparatus, one target portion at a time. In one type oflithographic apparatus, the pattern on the entire patterning device istransferred onto one target portion in one go; such an apparatus iscommonly referred to as a stepper. In an alternative apparatus, commonlyreferred to as a step-and-scan apparatus, a projection beam scans overthe patterning device in a given reference direction (the “scanning”direction) while synchronously moving the substrate parallel oranti-parallel to this reference direction. Different portions of thepattern on the patterning device are transferred to one target portionprogressively. Since, in general, the lithographic apparatus will have amagnification factor M (generally <1), the speed F at which thesubstrate is moved will be a factor M times that at which the projectionbeam scans the patterning device.

Prior to the device fabrication procedure of transferring the patternfrom the patterning device to the substrate of the device manufacturingprocess, the substrate may undergo various device fabrication proceduresof the device manufacturing process, such as priming, resist coating anda soft bake. After pattern transfer, the substrate may be subjected toother device fabrication procedures of the device manufacturing process,such as a post-exposure bake (PEB), development, a hard bake andmeasurement/inspection of the transferred pattern. This array of devicefabrication procedures is used as a basis to make an individual layer ofa device, e.g., an IC. The substrate may then undergo various devicefabrication procedures of the device manufacturing process such asetching, ion-implantation (doping), metallization, oxidation,chemo-mechanical polishing, metrology (e.g., using a scanning electronicmicroscope (SEM)), etc., all intended to finish off the individual layerof the device. If several layers are required in the device, then thewhole process, or a variant thereof, is repeated for each layer.Eventually, a device will be present in each target portion on thesubstrate. If there is a plurality of devices, these devices are thenseparated from one another by a technique such as dicing or sawing,whence the individual devices can be mounted on a carrier, connected topins, etc.

So, manufacturing devices, such as semiconductor devices, typicallyinvolves processing a substrate (e.g., a semiconductor wafer) using anumber of fabrication processes to form various features and multiplelayers of the devices. Such layers and features are typicallymanufactured and processed using, e.g., deposition, lithography, etch,chemical-mechanical polishing, and ion implantation. Multiple devicesmay be fabricated on a plurality of dies on a substrate and thenseparated into individual devices. This device manufacturing process maybe considered a patterning process. A patterning process involves apatterning step, such as optical or nanoimprint lithography using alithographic apparatus, to provide a pattern on a substrate andtypically, but optionally, involves one or more related patternprocessing steps, such as resist development by a development apparatus,baking of the substrate using a bake tool, etching using the patternusing an etch apparatus, etc. Further, one or more metrology processesare typically involved in the patterning process.

As semiconductor manufacturing processes continue to advance, thedimensions of functional elements have continually been reduced whilethe amount of functional elements, such as transistors, per device hasbeen steadily increasing over decades, following a trend commonlyreferred to as “Moore's law”. At the current state of technology, layersof devices are manufactured using lithographic projection apparatusesthat project a pattern corresponding to a design layout onto a substrateusing illumination from a deep-ultraviolet illumination source, creatingindividual functional elements having dimensions well below 100 nm, i.e.less than half the wavelength of the radiation from the illuminationsource (e.g., a 193 nm illumination source). This process in whichfeatures with dimensions smaller than the classical resolution limit ofa lithographic projection apparatus are printed, is commonly known aslow-k₁ lithography, according to the resolution formula CD=k₁×λ/NA,where λ is the wavelength of radiation employed (currently in most cases248 nm or 193 nm), NA is the numerical aperture of projection optics inthe lithographic projection apparatus, CD is the “critical dimension”—generally the smallest feature size printed—and k₁ is an empiricalresolution factor. In general, the smaller k₁ the more difficult itbecomes to reproduce a pattern on the substrate that resembles the shapeand dimensions planned by a circuit designer in order to achieveparticular electrical functionality and performance. To overcome thesedifficulties, sophisticated fine-tuning steps are applied to thelithographic projection apparatus and/or a pattern corresponding to adesign layout. These include, for example, but not limited to,optimization of NA and/or optical coherence settings, customizedillumination schemes, use of phase shifting patterning devices, opticalproximity correction (OPC) in the pattern corresponding to the designlayout (such as biasing of pattern feature, addition of an assistfeature, applying a serif to a pattern feature, etc.), or other methodsgenerally defined as “resolution enhancement techniques” (RET).

BRIEF SUMMARY

In an embodiment, there is provided a method for determining adeformation of a resist in a patterning process. The method includesobtaining a resist deformation model of a resist having a pattern, theresist deformation model configured to simulate a fluid flow of theresist due to capillary forces acting on a contour of at least onefeature of the pattern, and determining, via a processor and the resistdeformation model, a deformation of a resist pattern to be developedbased on an input pattern to the resist deformation model. The resistdeformation model is based on a linearized Navier-Stokes flow equations.The fluid flow is characterized by a Stokes flow, and/or a Hele-Shawflow.

Furthermore, in an embodiment, there is provided in a method fordetermining a parameter of a patterning process. The method includesobtaining (i) a patterning process model that includes a resistdeformation model of a resist having a pattern, the resist deformationmodel configured to simulate a fluid flow of the resist due to capillaryforces acting on a contour of at least one feature of the pattern, and(ii) a target pattern; determining, via a processor, a resist patternbased on a simulation of the patterning process model with the targetpattern as an input to the patterning process model, wherein adifference exists between the resist pattern and the target pattern; anddetermining, via the processor, a value of a parameter of the patterningprocess based on the simulation of the patterning process, the value ofthe parameter being determined such that the difference between theresist pattern and the target pattern is reduced. In an embodiment, theparameter of the patterning process comprises at least one of dose,focus, and optical proximity correction. The method further includesapplying the value of the parameter of the patterning process to alithographic apparatus during the patterning process.

Furthermore, in an embodiment, there is provided in a method fordetermining a deformation of a pattern to be formed in a patterningprocess. The method includes inputting, into a resist deformation model(e.g., a thin-film based model), pattern information relating to thepattern to be formed, the model configured to simulate deformation of aportion of a resist, the portion comprising a boundary liquid layerlocated at a boundary between a developed region in the resist and aregion of the resist surrounding the developed region, wherein the modelis configured to determine a first deformation component of the boundaryliquid layer caused by fluid flow of the boundary liquid layer and asecond deformation component of the boundary liquid layer caused by thefluid flow of the boundary liquid layer, and determining, via aprocessor, the deformation of the pattern to be formed in the resistbased on the input pattern information, wherein the deformationcomprises a combination of the first deformation component and thesecond deformation component of the boundary liquid layer. The boundaryliquid layer has a thickness smaller than a length of the developedregion in the resist at the boundary.

In an embodiment, the first deformation component is determined in ahorizontal plane based on a horizontal component of a flow rate of theboundary liquid layer and the second deformation is determined in thehorizontal plane based on a vertical component of the flow rate of theboundary liquid layer.

Furthermore, there is provided a method for determining a deformation ofa pattern to be formed in a patterning process. The method includesinputting, into a resist deformation model, pattern information relatingto the pattern to be formed, the model configured to simulatedeformation of a portion of a resist, the portion comprising a boundaryliquid layer located at a boundary between a developed region in theresist and a region of the resist surrounding the developed region,wherein the model is configured to determine a deformation of theboundary liquid layer caused by a horizontal fluid flow of the boundaryliquid layer; and determining, via a processor, the deformation of thepattern to be formed in the resist by simulating the resist deformationmodel based on the input pattern information. The boundary liquid layerhas a thickness smaller than a length of the developed region in theresist at the boundary.

Furthermore, there is provided a non-transitory computer program productcomprising machine-readable instructions for causing a processor tocause performance of the steps of the aforementioned method.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of the specification, illustrate one or more embodiments and,together with the description, explain these embodiments. Embodiments ofthe invention will now be described, by way of example only, withreference to the accompanying schematic drawings in which correspondingreference symbols indicate corresponding parts, and in which:

FIG. 1 is a block diagram of various subsystems of a lithography system,according to an embodiment.

FIG. 2 is a block diagram of simulation models of a computationallithography technique, according to an embodiment.

FIG. 3A illustrates a wafer image of a pattern having a plurality offeatures (e.g., bars) printed on the wafer, according to an embodiment.

FIG. 3B is the aerial image of the pattern printed on the wafer in FIG.3A, according to an embodiment.

FIG. 3C illustrates an example of contours obtained from the aerialimage of FIG. 3B, according to an embodiment.

FIG. 4A is an example flowchart for obtaining a resist deformation usinga resist deformation model, according to an embodiment.

FIG. 4B illustrates an example feature within a resist domain, accordingto an embodiment.

FIG. 4C, a force Fi at a vertex i determined based on neighboringvertices i−1 and i+1, respectively, according to an embodiment.

FIG. 4D illustrates resultant outflux in a vertical direction (e.g.,along y-axis) and influx in a horizontal direction (e.g., along x-axis)on the resist domain, according to an embodiment.

FIG. 4E illustrates an example boundary condition with rotlets R1, R2,R3, and R4 are placed at a boundary corner points around the feature,according to an embodiment.

FIG. 4F illustrates an example the boundary condition referred as a noflux condition, according to an embodiment.

FIG. 4G illustrates and example of a wall condition applied around aresist domain boundary, according to an embodiment.

FIG. 4H illustrates an example of periodic boundary conditions appliedaround a resist domain boundary, according to an embodiment.

FIG. 5 illustrates another example of determining resist deformation,according to an embodiment.

FIG. 6 illustrates a squeeze mode velocity field, according to anembodiment.

FIG. 7 illustrates examples of initial feature shapes and deformedfeatures corresponding to the initial features, according to anembodiment.

FIG. 8A is a flow chart of a method for determining resist deformation,according to an embodiment.

FIG. 8B is a flow chart of a method of simulation of the resistdeformation model used in FIG. 8A, according to an embodiment.

FIG. 9 is a flow chart of a method for determining a parameter of apatterning process based on a resist deformation computed based on theresist deformation model, according to an embodiment.

FIGS. 10A and 10B are a top cross-section view and side cross-sectionview, respectively, of a substrate having resist and a feature therein,according to an embodiment.

FIG. 10C illustrates a boundary liquid layer around a feature within theresist, according to an embodiment.

FIG. 11 shows a comparison between an initial contour, a deformedcontour based on a full-fluid resist deformation model, and a thin-filmbased resist deformation model, according to the present disclosure.

FIG. 12A illustrates example vertices and their movement according tothe boundary liquid layer of FIG. 10C based resist deformation model,according to the present disclosure.

FIG. 12B illustrates example symbols used in a vertical flowdetermination of the boundary layer of FIG. 10C, according to thepresent disclosure.

FIG. 13 illustrates example cross-section after shrinkage due to thevertical flow, according to the present disclosure.

FIG. 14A is flow chart of a method for determining a deformation of apattern to be developed using a boundary layer based resist deformationmodel, according to the present disclosure.

FIG. 14B is an example flow chart of a method for determining adeformation due to a horizontal flow of the boundary layer, according tothe present disclosure.

FIG. 15 illustrates example deformation based on the resist model usedin method of FIG. 14A.

FIG. 16 is a block diagram of an example computer system, according toan embodiment.

FIG. 17 is a schematic diagram of a lithography projection apparatus,according to an embodiment.

FIG. 18 is a schematic diagram of another lithography projectionapparatus, according to an embodiment.

FIG. 19 is a more detailed view of the apparatus of FIG. 18, accordingto an embodiment.

FIG. 20 is a more detailed view of the source collector module of theapparatus of FIG. 18 and FIG. 19, according to an embodiment.

DETAILED DESCRIPTION

As background to embodiments and turning to FIG. 1, there is illustratedan exemplary lithographic projection apparatus 10A. Major components area radiation source 12A, which may be a deep-ultraviolet excimer lasersource or other type of source including an extreme ultra violet (EUV)source, illumination optics which define the partial coherence (denotedas sigma) and which may include optics 14A, 16Aa and 16Ab that shaperadiation from the source 12A; a support configured to hold a patterningdevice 18A; and projection optics 16Ac that project an image of thepatterning device pattern onto a substrate plane 22A. An adjustablefilter or aperture 20A at the pupil plane of the projection optics mayrestrict the range of beam angles that impinge on the substrate plane22A, where the largest possible angle defines the numerical aperture ofthe projection optics NA=sin(Θ_(max)). In an embodiment, thelithographic projection apparatus itself need not have the radiationsource 12A.

So, in a lithographic projection apparatus, the optics 16Ac directs anaerial image of the patterning device pattern onto the substrate(typically a de-magnified version). An aerial image (AI) is theradiation intensity distribution at substrate level. A resist layer onthe substrate is exposed and the aerial image is transferred to theresist layer as a latent “resist image” (RI) therein. The resist image(RI) can be defined as a spatial distribution of solubility of theresist in the resist layer.

Now, it is often desirable to be able computationally determine how apatterning process would produce a desired pattern on a substrate. Thus,a simulation can be provided to simulate one or more parts of theprocess. For example, it is desirable to be able to simulate thelithography process of transferring the patterning device pattern onto aresist layer of a substrate as well as the yielded pattern in thatresist layer after development of the resist.

An exemplary flow chart for simulating lithography in a lithographicprojection apparatus is illustrated in FIG. 2. An illumination model 31represents optical characteristics (including radiation intensitydistribution and/or phase distribution) of the illumination. Aprojection optics model 32 represents optical characteristics (includingchanges to the radiation intensity distribution and/or the phasedistribution caused by the projection optics) of the projection optics.A design layout model 35 represents optical characteristics (includingchanges to the radiation intensity distribution and/or the phasedistribution caused by a given design layout) of a design layout, whichis the representation of an arrangement of features on or formed by apatterning device. An aerial image 36 can be simulated using theillumination model 31, the projection optics model 32 and the designlayout model 35. A resist image 38 can be simulated from the aerialimage 36 using a resist model 37. Simulation of lithography can, forexample, predict contours and/or CDs in the resist image.

More specifically, it is noted that the illumination model 31 canrepresent the optical characteristics of the illumination that include,but not limited to, NA-sigma (σ) settings as well as any particularillumination shape (e.g. off-axis illumination such as annular,quadrupole, dipole, etc.). The projection optics model 32 can representthe optical characteristics of the of the projection optics, include,for example, aberration, distortion, a refractive index, a physical sizeor dimension, etc. The design layout model 35 can also represent one ormore physical properties of a physical patterning device, as described,for example, in U.S. Pat. No. 7,587,704, which is incorporated byreference in its entirety. Optical properties associated with thelithographic projection apparatus (e.g., properties of the illumination,the patterning device and the projection optics) dictate the aerialimage. Since the patterning device used in the lithographic projectionapparatus can be changed, it is desirable to separate the opticalproperties of the patterning device from the optical properties of therest of the lithographic projection apparatus including at least theillumination and the projection optics, and hence design layout model35.

The resist model 37 can be used to calculate the resist image from theaerial image, an example of which can be found in U.S. Pat. No.8,200,468, which is hereby incorporated by reference in its entirety.The resist model is typically related only to properties of the resistlayer (e.g., effects of chemical processes which occur during exposure,post-exposure bake and/or development).

The objective of the simulation is to accurately predict, for example,edge placements, aerial image intensity slopes and/or CDs, which canthen be compared against an intended design. The intended design isgenerally defined as a pre-OPC design layout which can be provided in astandardized digital file format such as GDSII, OASIS or other fileformat.

From the design layout, one or more portions may be identified, whichare referred to as “clips”. In an embodiment, a set of clips isextracted, which represents the complicated patterns in the designlayout (typically about 50 to 1000 clips, although any number of clipsmay be used). As will be appreciated by those skilled in the art, thesepatterns or clips represent small portions (e.g., circuits, cells, etc.)of the design and especially the clips represent small portions forwhich particular attention and/or verification is needed. In otherwords, clips may be the portions of the design layout or may be similaror have a similar behavior of portions of the design layout wherecritical features are identified either by experience (including clipsprovided by a customer), by trial and error, or by running a full-chipsimulation. Clips often contain one or more test patterns or gaugepatterns. An initial larger set of clips may be provided a priori by acustomer based on known critical feature areas in a design layout whichrequire particular image optimization. Alternatively, in anotherembodiment, the initial larger set of clips may be extracted from theentire design layout by using some kind of automated (such as, machinevision) or manual algorithm that identifies the critical feature areas.

In some examples, the simulation and modeling can be used to configureone or more features of the patterning device pattern (e.g., performingoptical proximity correction), one or more features of the illumination(e.g., changing one or more characteristics of a spatial/angularintensity distribution of the illumination, such as change a shape),and/or one or more features of the projection optics (e.g., numericalaperture, etc.). Such configuration can be generally referred to as,respectively, mask optimization, source optimization and projectionoptimization. Such optimizations can be performed on their own orcombined in different combinations. One such example is source-maskoptimization (SMO) involves the configuring of one or more features ofthe patterning device pattern together with one or more features of theillumination. The optimization techniques may focus on one or more ofthe clips. The optimizations may use the simulations described herein toproduce values of various parameters.

In an optimization process of a system, a figure of merit of the systemcan be represented as a cost function. The optimization process boilsdown to a process of finding a set of parameters (design variables) ofthe system that minimizes the cost function. The cost function can haveany suitable form depending on the goal of the optimization. Forexample, the cost function can be weighted root mean square (RMS) ofdeviations of certain characteristics (evaluation points) of the systemwith respect to the intended values (e.g., ideal values) of thesecharacteristics; the cost function can also be the maximum of thesedeviations (i.e., worst deviation). The term “evaluation points” hereinshould be interpreted broadly to include any characteristics of thesystem. The design variables of the system can be confined to finiteranges and/or be interdependent due to practicalities of implementationsof the system. In case of a lithographic projection apparatus, theconstraints are often associated with physical properties andcharacteristics of the hardware such as tunable ranges, and/orpatterning device manufacturability design rules and the evaluationpoints can include physical points on a resist image on a substrate, aswell as non-physical characteristics such as dose and focus.

As noted above, a layer on a substrate can have pattern transferredthereto. Such a layer will be generally referred to as a resist layerand may have various chemical compositions. In an embodiment, the resistlayer is a layer of radiation-sensitive material. The resist layerusually has a small but finite thickness that may be comparable in sizeto patterns imaged onto the resist. The resist layer may undergo varioustreatments in a lithographic process. For example, the resist may beexposed to radiation such as EUV or DUV, which induces chemicalreactions in the resist. The resist may undergo a post-exposure bake(PEB), development (e.g., positive tone development or negative tonedevelopment), and/or a hard bake. Each of these treatments may cause theresist to deform in one, two or three dimensions and the deformation maybe location dependent (e.g., three-dimensional location dependent). Thedeformation of the resist may affect downstream treatments such asmaterial deposition and etching. In a lithographic process usingnegative tone development, the impact of the resist deformation onresist top loss and critical dimension may be especially significant.Therefore, a resist model 37 with the ability to predict deformation ofthe resist is beneficial to more accurate lithography and higher yield.The resist model 37 may also be able to predict the reaction of theresist layer to various other physical and chemical treatments in thelithographic process. An exemplary resist model according to an aspectof the present disclosure is described later.

A pattern is formed in the resist layer, e.g., by exposing the resistlayer to radiation via a patterning device. The resist layer may undergopost-exposure treatments such as PEB and deform into a deformed resistlayer with deformed pattern therein. In an embodiment, a portion of thedeformed resist layer that received sufficiently high dose during theexposure remain on the substrate after negative tone development and therest (e.g., portions) of the deformed resist layer is dissolved afternegative tone development. Alternatively, the portion of the deformedresist layer received sufficiently high dose during the exposure todissolve during positive tone development and the rest of the deformedresist layer remains on the substrate after positive tone development.Whether the portion remains or dissolves depends on the chemicalcomposition of the resist and the chemical composition of the developer.

As noted above, simulation models attempt to accurately predict patternsin a resist profile. The resist is a viscoelastic material and, for aperiod of time, the resist may exhibit a fluidic behavior that issignificant. Under this assumption, intermolecular forces, pressureand/or other forces (generally referred to herein as intrafluid forces)can result in a stress in the fluid that yields a viscous flow.Simulation models fail to account for such a viscous flow within theresist. Moreover, the effects of intrafluid forces acting on the resisttend to be significant when the resist pattern is dense. So, simulationmodels are not able to predict, with high accuracy, shapes of a resistpattern (particularly dense patterns or patterns with high curvature) inresist, which can lead to downstream effects such as modeling errors.

The resist pattern includes a plurality of developed portions that aresurrounded by resist. One or more intrafluid forces, (e.g., surfacetension) typically act on the resist thereby deforming the resist at theboundaries of the developed portions of the resist. Thus, a shape of aresist pattern, corresponding to a developed portion, deforms at severallocations along the boundary of the developed portion. In an embodiment,a resist deformation model is developed under the assumption thatcapillary and viscous flow effects are dominant. Furthermore, the resistdeformation model is developed under the assumption that continuumdescription holds. In an embodiment, the driving force acting on theresist is a capillary force that acts at an interface between twomaterials such as the resist and an inner portion (e.g., air) of thedeveloped portion, e.g., as discussed with respect to contours in image320 of FIG. 3C below. In an embodiment, the contour is obtained vialithographic simulations and contour extraction algorithms known to aperson skilled in the art.

In an embodiment, a mask pattern layout of a pattern to be printed onthe wafer is obtained. From the mask pattern layout, an aerial image(e.g., in FIG. 3B) is generated, for example, via lithographicsimulation software such as Tachyon, HyperLith, or ProLith or othersoftware configured to generate an aerial image from the mask patternlayout. In a typical situation, the aerial image is a grayscale image,wherein each pixel of the image corresponds to a different radiationintensity (i.e., before the radiation impinges on the resist). Anexample of an aerial image (e.g., 310) and corresponding pattern (e.g.,300) printed on a wafer is illustrated in FIGS. 3B and 3A, respectively.

FIG. 3A illustrates a wafer image 300 of a pattern having a plurality offeatures (e.g., bars) printed on the wafer. It can be seen that contoursof the features are deformed due to deformation of the resist on thewafer. It is beneficial to predict deformation of such features of thepattern in the resist to adjust or control parameters of the patterningprocess so that desired patterns are printed without defects, therebyincreasing yield of the patterning process. In an embodiment, suchdeformation in the resist is predicted based on the resist deformationmodel (e.g., a fluid dynamics model) discussed in the present disclosurebelow, e.g., with respect to FIG. 4A and FIG. 5.

FIG. 3B is the aerial image 310 of the pattern printed on the wafer (inFIG. 3A). In an embodiment, the aerial image 310 is a grayscale imageand each pixel corresponds to a different light intensity (before thelight hits the resist). From the aerial image 310, contours of thepattern can be extracted at some threshold level. For example,extracting contours of the aerial image at an intensity value breachinga desired threshold.

FIG. 3C illustrates an example of an image 320 including contours in aresist 323 (also referred as resist contours) obtained from the aerialimage 310 that can be input to the deformation model of the presentdisclosure to determine deformation of the resist. In an embodiment, theimage 320 including the contours is a binary image. The image 320includes a plurality of contours of features (e.g., bars, contact hole,etc.) of a pattern surrounded by resist. A binary image represents theoutline of the each of the features of the pattern.

In an embodiment, in the contours 320, everything inside the extractedcontours is gas 321 (white area) and outside these contours is theresist 323 (grey area). Then, both the resist 323 and the gas 321 can besimulated as different or the same fluids to determine the deformationof the contours. When using Stokeslet or Hele-Shaw flow, the fluids 323and 321 are considered to have the same viscosity. In an embodiment, thesimulation is based on a 2D resist deformation model, where a resistheight or thickness is not taken into account. However, the presentdisclosure is not limited to 2D model, and can be modified to apply to a3D resist as well. An example of a resist deformation process isexplained with respect to FIG. 4A.

FIG. 4A is an example flowchart for obtaining a resist deformation usinga resist deformation model, where the resist deformation model is afluid dynamics model based on Navier-Stokes equations that capture aviscous flow in the resist arising from one or more intrafluid forcesand computes a corresponding deformation of the resist. So, the fluiddynamics model is able to accurately predict, e.g., complex shapes ofthe resist pattern. In an embodiment, the fluid dynamics model islinearized, for example, Stokes flow (e.g., a 2D Stokes flow)represented by a linearized Navier-Stokes equation.

In an embodiment, obtaining resist deformation involves simulation ofthe fluid dynamics model using an image comprising a contour(s) of afeature(s) as input. In an embodiment, the contours are extracted (e.g.,in 320) from an aerial image (e.g., in 310) as discussed with respect toFIGS. 3B and 3C above. The extracted contours can be a grey scale imageor a binary image. In an embodiment, the extracted contour refers topoints where the force due to surface tension will be applied.

On the extracted contours, forces (also referred as capillary forces) atdifferent location along the contours are determined resulting in aforce field. The forces or force field along the contours causedeformation of the features. In an embodiment, forces acting on otherfeatures may cause deformation of the desired feature. The forces arefurther used to determine a velocity field (or flow field) at and aroundthe contours of the features of the resist. In an embodiment, thevelocity field mimics the movement of the edges of the contours overtime that eventually results in a final deformation of the contour ofthe feature in the resist.

In an embodiment, the simulation is an iterative process. In aniteration, a first step P41 involves determining a force (F_(i)) at alocation (i) along the desired contour. In an embodiment, the force iscalculated for the location (i) along the desired contour (e.g., asillustrated in FIG. 4B) based on the forces at neighboring locations(j), where j is different from i. Such force (F_(i)) is calculated foreach location along the contour based on other locations along thecontour. In an embodiment, the location (i) and/or (j) is represented bya vertex or a point. The equation and approach used to determine a forceis explained later in the disclosure (e.g., equations 3-5). In anembodiment, the force may be calculated using Stokes' equation, which isa linearized version of a Navier-Stokes equation obtained by neglectinginertial terms, as given below (equation 1).

−∇P+μ∇ ² {right arrow over (u)}+{right arrow over (F)}=0  (1)

In equation 1, the pressure P is such that the divergence of thevelocity field vanishes in the resist domain. The velocity {right arrowover (u)} is thus completely determined by the force {right arrow over(F)} on the liquid. This equation is linear. Thus, the flow field(interchangeably referred as velocity or velocity flow field) is aconvolution of a response to a point force, and the applied forces. Inan embodiment, velocity field is driven by capillary forces determinedusing equation (3-5), discussed later.

Once the forces at each location along the contour is determined, asecond step P43 involves determining a total velocity field (u) by theforce (F_(i)) at all other points than point i. In an embodiment, thevelocity field at a point j is determined by evaluating equationu_(j)=ΣG_(ij)·F_(i)∨i,j+regularization, where G_(jj)=0 (Einstein'ssummation convention is used in notation herein). In an embodiment, thefluid dynamic model may require regularization, as solving for velocityfield using the fluid dynamics model may be an ill-posed problem thatmay result in unstable simulation (e.g., due to singularity that mayoccur at a vertex or a point along the contour). Thus, regularizationmay provide a stability in simulation. In an embodiment, theregularization may be done to solve problem such as the singularbehavior of the Stokeslet. Such singular behavior disappears when theapplied force is distributed over a line segment, instead of beingapplied at a single point, which is equivalent to applying a pair ofrotlets of equal magnitude and opposite sign at the ends of the linesegment (e.g., see FIG. 4E).

According to an embodiment, a velocity response G₀(x−{tilde over (x)})at position x to a point force at position {tilde over (x)} is called aStokeslet, given by equation (2) below:

$\begin{matrix}{{G_{ij}(x)} = {{{- \delta_{ij}}{\ln\left( \sqrt{x_{k}x_{k}} \right)}} + \frac{x_{i}x_{j}}{x_{k}x_{k}}}} & (2)\end{matrix}$

In an embodiment, the flow may be a 2D Stokes flow and G (inaforementioned equation) represents Green's function. The equation andapproach (e.g., using Stokeslets and boundary conditions) used todetermine a velocity is explained later in the disclosure. In anembodiment, boundary conditions on the resist or resist domain may beimplemented in different ways, for example, as illustrated in FIGS.4D-4F and explained in detail later in the present disclosure. Theseboundary conditions are applied to the resist domain, which enabledetermination of the resist deformation, when the equations such as 1and 2 are simulated.

In an embodiment, the iteration involves a third step P45, where avelocity at point i is be determined. In an embodiment, the velocity atpoint i is determined by the equation Σ(r_(i))n_(i)·u_(i)=0, where r isa the distance between the neighboring points, x_(i−1) and x_(i+1), ofpoint i, n is the normal vector, and u_(i) is a velocity at point i. Inan embodiment, this step is applied to conserve the surface area of thedesired polygon or any other polygon for that matter.

Further, the first P41, second P43 and/or the third P45 steps arerepeated and the velocity fields at all locations along the contour andcorrections corresponding to each such location are superimposed todetermine a total velocity field at time step t_(n). Further, in stepP47, a position differential may be equated to a velocity to determine afinal deformation. For example, a differential equation

$\frac{{dx}_{i}}{dt} = u_{i}$

may be solve for position x_(i) to determine a final deformation of thedesired feature or resist in general. For example, the differentialequation may be solved using an ODE solving methods such as Eulerexplicit method which would require small time steps to obtain accuratesolutions or use higher order Runge-Kutta methods.

An example of the process for determining resist deformation and relatedconditions enabling simulation of the resist deformation model isexplained for a desired feature 452 with respect to FIGS. 4B-4F.

FIG. 4B illustrates an example feature 452 within a resist domain 450.The feature 452 is interchangeably referred as a contour 452. Asmentioned earlier, the contour 452 may be extracted from a correspondingaerial image. In an embodiment, the contour 452 is divided into verticesor points such as 452 a, 452 b, 452 c, 452 d, and so on representing aset of locations within the resist domain 450. Such vertices or points(e.g., 452 a-452 d) may be arranged or distributed along the contour 452such that the area within the contour 452 is conserved. Forces andvelocities at such vertices are further computed.

In an embodiment, referring to FIG. 4C, a force Fi at a vertex idetermined based on vertex i and neighboring vertices i−1 and i+1respectively. In an embodiment, the force Fi is computed based on aninterfacial tension (σ) of an interface between the two liquids, and ontangential vectors τ before and after the vertex i. An example equationused to compute Fi is as follows:

$\begin{matrix}{{\overset{\rightarrow}{F}}_{i} = {\sigma\left( {{\overset{\rightarrow}{\tau}}_{i + \frac{1}{2}} - {\overset{\rightarrow}{\tau}}_{i - \frac{1}{2}}} \right)}} & (3) \\{{\overset{\rightarrow}{\tau}}_{i + \frac{1}{2}} = \frac{{\overset{\rightarrow}{r}}_{i + \frac{1}{2}}}{{\overset{\rightarrow}{r}}_{i + \frac{1}{2}}}} & (4) \\{{\overset{\rightarrow}{r}}_{i + \frac{1}{2}} = {{\overset{\rightarrow}{x}}_{i + 1} + {- {\overset{\rightarrow}{x}}_{i}}}} & (5)\end{matrix}$

In equation 3, σ is surface tension at the interface (i.e., at thecontour of the feature) which is a property of the resist used forprinting a pattern on a substrate. The tangential vectors are computedusing equations 4-5. Once, the forces are determined, the velocityresulting from the forces is determined based on, for example,Stokeslets. Further, the velocity determination involves assigningcertain boundary conditions on the resist domain, for example, boundaryconditions illustrates with respect to FIGS. 4D-4F.

In an embodiment, force points are defined along a wall outside a resistdomain to generate an additional velocity field. In an embodiment, avalue of a force at each force point is such that the total superposedvelocity field adds up to zero velocity at the boundary points (e.g., noflux) or equals velocity for boundary nodes at opposite locations. Allforce points (e.g., along the wall) together give a set of equationswith equal number of unknowns. In an embodiment, a number of forcepoints is approximately (or the same, in an embodiment) to number ofvelocity nodes at the boundary of the resist domain. In an embodiment,locations of the force points are ideally close but not identical to thevelocity node. Spacing of the force points is less than or equal to aminimum distance of a feature to a wall.

In an embodiment, the individual forces along the contour 452 of theresist results in an effective forces acting at the resist domainboundary (e.g., 450) that causes a squeeze mode in the resist domain.Such squeeze mode refers to deformation or movement of the desiredfeatures due to effective forces from adjacent features. In anembodiment, the squeeze mode is eliminated by applying one or moreboundary conditions, as discussed with respect to FIGS. 4D-4F.

In an embodiment, FIG. 4D illustrates resultant influx in a horizontaldirection (e.g., along x-axis) and outflux in a vertical direction(e.g., along y-axis) on the resist domain 450. In an embodiment, theresist domain 450 is considered an infinite domain for simulationpurposes. Further, no boundary conditions are defined. As such, multiplefeatures together could result in an overall squeeze mode flow. Toeliminate the squeeze mode and retain only a motion of interest (e.g.,motion of the feature 452), rotlets R1, R2, R3, and R4 are placed at aboundary corner points around the feature 452, as shown in FIG. 4E. Therotlets R1-R4 produce a flow counter to the one created because offorces along the feature 452. In an example, to counteract, the drawnrotlets R1-R4 in FIG. 4E changes direction (i.e. clockwise changes intoanti-clockwise and viceversa).

In an embodiment, the squeeze mode is eliminated by applying rotlets ofstrength S defined based on making a first moment of all the forces atthe vertices of the contour to zero. For example, using equationS=Σ(F_(xn)·x_(n)−F_(yn)·Y_(n)), where Fxn and Fyn are forces in x and ydirections, respectively, n refers to a number of point (e.g., points450 a, 450 b, 450 c, 450 d, etc.) along the resist domain 450 at whichthe force is determined, and xn and Yn refers to the position of then^(th)-node.

In another embodiment, referring to FIG. 4F, the boundary condition maybe a no flux condition achieved by adding n force points 460 a, 460 b,460 c, etc. on an outside location 460, i.e., at a certain distance fromthe domain 450 to cancel an outflux and/or influx on the n boundarypoints 450 a, 450 b, 450 c, 450 d, etc., respectively, to impose a noflux condition. The n boundary points 450 a, 450 b, 450 c, 450 d, etc.may also be referred as velocity nodes. Velocity nodes are nodes on thedomain boundary at which a velocity field is determined. It is at thesevelocity nodes, where a boundary condition may be applied.

In some cases, either periodic boundary conditions or wall boundaryconditions are needed. In an embodiment, implementing such boundaryconditions in a singularity method (e.g., if Green function's issingular in x=0) involves use of fundamental solutions that satisfy theboundary conditions. Unfortunately, deriving the fundamental solutionfor 2D Stokes flow in a periodic domain is a significant effort andclosed form solutions are not found in the state-of-the-art. Instead ofusing a fundamental solution that satisfies the boundary conditions, thepresent methods impose boundary conditions directly.

FIG. 4G illustrates an example of a wall condition. The figure shows thevelocity field due to forces along the contours of the features (orfeature in general). At a wall 475 (outside the resist domain), a liquid(e.g., resist within boundary 470) adjacent to the wall 475 moves withthe same velocity as the wall. If the wall 475 is stationary, bothcomponents of the liquid velocity at the wall 475 are zero. The wall 475keeps the velocity of the liquid 470 equal to zero by applying a forceon the liquid 470. In proposed method, a force is applied on the liquidalong its entire length, keeping the velocity zero along its entirelength. In an embodiment, the force is applied at a finite number ofpoints 475 a, 475 b, 475 c, etc. (also referred as nodes) along the wall475. This also implies that the velocity can be set to zero at a finitenumber of points. These points cannot coincide with the points where theforce is applied, since the fundamental solution is singular there,unless we use some regularization. In an embodiment, the nodes (similarto that in FIG. 4F) are placed at a domain boundary 470, and the forcepoints 475 a, 475 b, 475 c, etc. are placed outside of the domain havinga distance between force points that is approximately similar to thespacing of the nodes of the domain boundary 470.

In an embodiment, the number of domain boundary points (or nodes) shouldbe large enough to make a truncation error (due to approximations in themodel) small enough, but small enough that the calculation time isacceptable. In an embodiment, the spacing between the nodes is not largewith respect to the smallest distance of the features to the boundary.In an embodiment, the spacing is equal to the smallest distance of thefeatures to the boundary. Based on such wall condition, the resultingdeformation and velocity fields for 2D Stokes flow with wall boundaryconditions are illustrated in FIG. 4G. The forces are applied along thewall 475 at the locations marked by dots (e.g., 475 a, 475 b, 475 c,etc.) outside the domain boundary 470. These forces are such that thevelocity at the dots along the domain boundary 470 is zero.

In an embodiment, illustrated in FIG. 4H, periodic boundary conditionsare applied around the domain boundary. The figure shows the velocityfield due to forces along the contours of the features (or feature ingeneral). Periodic boundary conditions have additional constraints. Thefirst constraint is that the velocity at opposing sides of the domainmust be equal. The second constraint is that the stress on a boundarymust be opposite to the stress on the opposing side. In an embodiment,such periodic constraint is imposed as pair of equal and opposite forcesat any point and its connected point. This ensures that the total forceis zero. Since the forces that are to be countered exert no torque, theimposed forces will also have zero torque. If a torque is exertedsomewhere in the domain, the periodic boundary conditions algorithm willcounter it. The sum of the forces in any pair is zero. So, a differencein a pair of forces is determined. The criterion for the magnitude ofthis force difference is that the velocity difference at thecorresponding nodes of the resist domain must vanish. The velocity at anode due to a pair of forces is obtained by evaluating the difference offundamental solutions. The requirement that at every node, the sum ofthese velocities must be opposite to the velocities due to the features,constitutes a system of linear equations, which can be solved usingscientific software such as Matlab that is configured to solve a set oflinear equation, differential equations, or other mathematicalcomputation.

In an embodiment, the resulting deformation and velocity fields for 2DStokes flow with periodic boundary conditions is calculated. The forcesare applied at the locations (dots) marked along outside boundary 485.These forces are such that the velocity at each dot along the domainboundary 480 is equal to the velocity at the opposing location on 480.

FIG. 5 illustrates another example of determining resist deformation. Asdiscussed earlier in FIGS. 3C and 4A, contours or extracted contours ofthe features from the aerial image are converted to polygons. Thepolygons represent a shape of the contours and may be constrained suchthat the area of the polygon is preserved. In an embodiment, the polygonincludes boundary or edges that are represented by vertices (e.g.,represented by dots) in an input 500. In an embodiment, a plurality ofvertices are associated with position information along the contours.For example, as shown, the input 500 comprises vertices represented bydotted lines (e.g., 501) of a feature, where each vertex is associatedwith position information within the input 500. The position of a vertex(e.g., of 501) is represented in terms of Cartesian co-ordinates, polarco-ordinates, relative position with respect to another feature, etc.

The input 500 may include a plurality of features (e.g., bars, holes,lines, etc.) which are converted to polygon or vertices representingpolygons. A plurality of vertices of a feature and a location of eachvertex thereby represents a geometry of the contour of the feature. Inan embodiment, the plurality of vertices are redistributed to make themapproximately evenly (or uniformly) spaced while conserving an area (orvolume in case of 3D resist deformation process) of the geometry of eachcontour. Thus, in an embodiment, each feature e.g., 501, 503, 505, etc.of the input 500 may be converted into a plurality of vertices andfurther the vertices may be redistributed or constrained such that thearea of the respective feature or contour representing the feature isconserved.

According to an embodiment, redistribution of the plurality of verticesmay be desired to maintain stability during simulation of the resistdeformation, as the simulation involves computation of matrix betweenforce and velocity that converts a force to a velocity using, forexample, Stokeslets (equation 2, discussed above). For example, theStokeslet does not have to be evaluated at a source vertex, because thevelocity at the vertex can be calculated from continuity once thevelocity at the other vertices is known. However, such approach resultsin truncation errors being collected into the velocity of a vertex. Forsome vertices, this results in a force to velocity matrix that is notdiagonally dominant, leading to an unstable scheme when the velocitiesare used to advance the positions. Thus, in an embodiment,redistributing the vertices to make the spacing even, may be sufficientto maintain stability in simulation of the resist deformation.

Once the input 500 is obtained, the fluid dynamics of the resist isdetermined in terms of forces and velocity that represent movement ofthe contour of the features in the resist. In an embodiment, the forcesor a force field are computed on a capillary surface, such as a contourof the feature or polygon and a movement of the plurality of verticesalong the contours is tracked.

In an embodiment, a flow of a body of liquid (e.g., resist) isconsidered to be a 2D Stokes flow, driven by capillary forces in compactarea or a selected area (also referred as a resist domain). According toan embodiment, the force on each vertex of the contour is a sum of thetensions on either side of the vertex. This force is exerted alongdifferent location on contour of the liquid, so Stokeslet per vertex isdetermined. The strength of the Stokeslet depends on an interfacialtension (σ) of an interface between the two liquids, and on tangentialvectors before and after the vertex, as discussed in FIG. 4A, withrespect to equations 3-5. In an embodiment, the curvature can becalculated not only between two neighbors per vertex but from moreneighboring vertices In an embodiment, such higher order differencingschemes for the calculation of the capillary forces, i.e., curvaturedetermination based on more than two neighboring vertices are possible.However, such higher ordering differencing schemes make it (too)difficult to determine a scheme that conserves volume and is stableduring simulation. Also, the formulation becomes much more complicated,increasing the chance of errors. Existing methods did not use the loworder method described herein, is as it was believed to be not possible.

In an embodiment, the equations 3-5 are applied to each feature (e.g.,501, 503, 505) and vertices corresponding to the feature of the input500. Thereby, the input 500 is transformed into a velocity field at agiven time step. For example, the velocity field 510 at a first timestep t1 is represented. In the velocity field 510, flow vectors 513,515, 511 are obtained in and around the features (e.g., 503 and 505).

In an embodiment, the forces are applied to the liquid (e.g., theresist) by taking these forces as the coefficients of the Stokeslets,and thus find the velocity anywhere except for the locations of thevertices. The locations of the vertices are excluded because theStokeslet G({right arrow over (x)}_(i)) is singular at {right arrow over(x)}={right arrow over (x)}_(i), at the location of the vertex. Toobtain the velocities of a vertex, the flow due to the force at thatvertex requires special treatment. The flow due to the forces at all theother vertices requires no special treatment. For example, to calculatethe velocity at a vertex, only the flow field due to that same vertexneeds to be regularized.

Upon further simulation, a velocity field 520 at a second time step t2is obtained further to the velocity field 510. It can be seen that asthe flow progresses, the feature such as 513 (or 515) deform into todeformed feature 523 (or 525). The deformed feature 523 (or 525) appearto be curved or circular compared to the original elongated shape of thefeature 503 (or 505) of the input 500. However, as mentioned earlier,constraints applied to the resist domain and the boundary conditionspreserves the area of the feature 503 (or 505) upon deformation of thefeature. Thus, areas of the deformed features 523 (or 525) and features503 (or 505) of the input 500 are approximately similar, however, theshape may change substantially. The velocity fields 510 and 520 anddeformation of the features shown in FIG. 5 are simply an instance ofthe simulation of the resist deformation model. For example, simulationof set of equations (1-5) at different vertices along with the boundaryconditions as discussed above. In an embodiment, a different instance ofdeformation may be obtained based on the instance of time for whichdeformed feature must be determined. For example, the simulation may berun for 10 s, then the result is a deformed resist and correspondingfeatures at 10 s. Similarly, the simulation may be run for 20 s, 50 s,etc. to obtain different deformation instances.

As mentioned earlier, during the simulation process, a problem of thesingular behavior of the Stokeslet disappears when the applied force isdistributed over a line segment, instead of being applied at a singlepoint. In an embodiment, the regularization of velocity field (e.g.,based on Stokeslet) may be performed, for example, based on followingequation (6).

$\begin{matrix}{{\overset{\rightarrow}{u}\left( \overset{\rightarrow}{x} \right)} = {- {\overset{\rightarrow}{F}\left( {{\ln\left( {\frac{1}{2}{{\overset{\rightarrow}{r}}_{i}}} \right)} - 1} \right)}}} & (6)\end{matrix}$

Equation 6 gives the velocity at the center of a line segment due to aforce F that is distributed over the line segment with a constant forcedensity. This velocity is finite for any line segment of finite length,which allows the velocity to be evaluated everywhere.

As mentioned earlier, a polygon (also called contour or feature) in theresist is an approximation of the real interface shape. The presence ofsharp corners in this representation is what causes the capillary forcesto be concentrated in a finite number of points. In an embodiment, awhole interface is curved so the force is applied over the entireinterface. When one vertex is considered, the distribution of the forcesover some far away region is relatively less important. How the forcesare distributed becomes relevant when the size of that region iscomparable to the distance of that region. At a bare minimum, the forcedue to the sections of interface directly adjacent to the vertex must bedistributed over a line segment as calculated above.

In another embodiment, the Stokeslet may not be evaluated at the sourcevertex, because the velocity at the vertex can be calculated fromcontinuity once the velocity at the other vertices is known. To eachvertex, a curve length |r_(i)| is assigned and normal vectors n_(x) andn_(y) along x and y axis, respectively, are assigned according toequations 7 and 8 as follows:

n _(x) =r _(y)  (7)

n _(y) =−r _(x)  (8)

By continuity and the divergence theorem, the total outward flux shouldbe zero, which can be computed based on equation (9) below)

$\begin{matrix}{{\sum\limits_{i}{{{\overset{\rightarrow}{r}}_{i}}\mspace{20mu}{{\overset{\rightarrow}{n}}_{i} \cdot {\overset{\rightarrow}{u}}_{i}}}} = 0} & (9)\end{matrix}$

With above equation (9), a task of regularization involves determiningu_(j) when u_(i≠j) are known, such that the continuity equation issatisfied. In other words, velocity is determined at all other vertices,except at i. Equation (9) yields the required velocity directly.Further, with the continuity based regularization, redistributing thepoints to make the spacing even maintains stability during simulation.

According to an embodiment, a limitation of 2D Stokes flow as a modelfor resist deformation is domain size matters. For a given forcing, thevelocity field depends on the location of the wall around the resistdomain. In an infinite domain, the velocity diverges. So, in anembodiment, the mitigation for velocity divergence is to only considerforce distributions that sum to zero. So, the flow field due to adistribution of features with capillary forces that spans a large partof space is calculated. The contribution to the velocity field due tofeatures that are far away, should be small. This requires a velocityfield that falls off as r⁻². As such, the velocity field due to thecapillary forces on the features is decomposed. In an embodiment, thevelocity field is decomposed into a squeeze flow (see FIG. 7) and a partthat falls off as r⁻² or faster. The squeeze mode velocity field isdriven by the diagonal part of {right arrow over (τ)}, the moment offorce. In an embodiment, in order to remove the squeeze mode, aconstraint is defined such that the sum of the diagonal part of themoment of force of the Stokeslets, and of the squeeze mode flow field isadded to zero.

FIG. 6 illustrates a squeeze mode velocity field 600. This velocityfield was calculated by summing Stokeslets at the vertical domainboundaries. The force per unit length on the boundary on the left isconstant, and opposite to the force per unit length in the boundary onthe right, which is also constant.

In a different approach, as mentioned earlier with respect to FIG. 4F, azero total flux condition is imposed through the horizontal walls of theresist domain, and thus zero flux through the vertical walls, by addingan extensional simple flow.

In the direct implementation of the Stokeslet based calculation, everyboundary point directly influences every other boundary point.Therefore, in the limit of a large number of boundary points N, therequired number of floating point operations (FLOPS) to calculate thevelocity field is quadratic (i.e., N²).

In an embodiment, to improve scaling in the Stokeslet based calculation,the velocity field due to far away features is replaced with theirtruncated multipole expansion. In the simplest form of the multipoleexpansion, only the squeeze mode is retained and all higher order termsare neglected. This is because the influence of all the squeeze modeterms does not converge as more distant terms are included, while thehigher order terms due to features at a distance between R and R+dR goesto zero as R increases. The squeeze mode flow due to each distancefeature is calculated, and each feature is replaced by a combination of4 rotlets as described earlier. These rotlets are placed at the cornersof a rectangle that is not large with respect to the replaced feature,such as the bounding rectangle of the feature. Thus, for each featurewhose influence is simplified in this way, the number of points in theglobal calculation N is reduced by n−4, at a cost of summing n forces.This is a significant improvement since the computational cost in theglobal calculation scales with N{circumflex over ( )}2. Thus, in anembodiment, the scaling of the velocity calculation with n points perfeature and m features is improved from n²m² to n²+nm². For example,assuming that n=10³ and m=10⁹, this implies a 10³ fold improvement,which is significant. In an embodiment, the scaling can be furtherimproved by collecting the multipole expansions of features inrelatively far away regions (e.g., using a fast multipole method).

Furthermore, in an embodiment, a further improvement in calculation timecan be obtained by neglecting even more distant features entirely. Inthis approximation, there are only local interactions. The distance atwhich interactions are neglected, can be specified as a tolerance on therelative magnitude. As relative magnitude, a ratio of velocity at agiven distance may be taken over the velocity at a distance equal to thefeature size. For instance, if a tolerance of 14% relative error due tothis approximation is set, and a tolerance of 5% relative error due tothe point spacing is set, the total relative error is below 20%. The cutoff distance for each feature is calculated as the distance where therelative magnitude of the multipole expansion of that feature dropsbelow the tolerance. This gives a calculation time scaling that islinear in the number of features and in the number of points perfeature. In other words, FLOPS is approximately nm.

After the velocity fields are computed, a final deformation isdetermined at a desired location. The desired location may be any pointin the resist, a point on a contour of a desired feature, or a pluralityof points on a desired contour. In an embodiment, the final position atany time “t” is determined based on integration of the velocity field ator around the desired location. For example, the final displacement maybe determined using a displacement equation 11 as follows:

{right arrow over (x _(i))}(t)=∫_(t) _(e) ^(t) {right arrow over(u)}({right arrow over (x _(i))}(t))dt  (11)

In above equation, {right arrow over (x_(i))}(t) is the displacement apoint i at time t, and u is the velocity field determined, for example,using Stokeslets, as discussed earlier. In an embodiment, the velocityfield {right arrow over (u)} is determined between times t₀ (i.e., astart time) and t_(e) (i.e., an end time of simulation). In anembodiment, the input (e.g., in FIG. 4A, 500), the velocity field (e.g.,510, 520) may be represented as pixelated images. Thus, in anembodiment, pixel values at time t₀ at any position may be determined byinterpolation. The pixel values should be constant along a curve inspace-time that is traced out by points moving with the fluid (e.g., theresist). Thus, a point i may correspond to a pixel value anddisplacement of the pixel correspond to displacement of the resist.

In an embodiment, feature shapes are obtained from an image by a contourfinding algorithm. In an embodiment, changing a value of the contour maybe desired without redoing the force and velocity calculations. Thus, inan embodiment, an input image may be calculated after deformationaccording to the 2D Stokes flow. Such reverse computation of an inputimage from deformation also allows further analyses that may be needed.For example, an option of calculating only a part of the image, such asan immediate neighborhood of the deformed contour, may be desired, asillustrated in FIG. 7.

In FIG. 7, initial feature shapes 703 and 705 (rounded elongatedrectangular features) are shown for reference, and deformed features 713and 715 correspond to the initial features 703 and 705. The deformedfeatures 713 and 715 are obtained from simulation of forces andvelocities corresponding to the initial features 703 and 705 based onthe resist deformation model and the simulation process discussed withrespect to FIGS. 4A and 5. In an embodiment, not only deformation of theinitial features 703 and 705 is determined, but also deformation of animmediate neighborhood of the resist represented by several instances of721 can be determined, for example, based on forces and velocitiesacting on the contour of the deformed features 713 and 715.

In an embodiment, the deformation of the entire resist domain may beobtained, for example, using position information and the displacementequation 11 above. Based on the equations (e.g., 2-11), the deformationmay be determined for each pixel of the velocity field, therebyresulting in deformation of an entire resist domain.

It can be understood by a person skilled in the art that the abovemethods and examples were explained with respect to 2D Stokes flow forconveying the concepts. However, the above methods are not limited to 2DStokes flow, and any other flow may be used to represent fluid dynamicsmodel and related boundary conditions may be applied to determine forcesand velocity field. In embodiment, the 2D Stokes flow driven bycapillary forces may be local in nature to a desired feature, so for anyrequired accuracy, there is a distance at which the velocity field maybe neglected due to feature edges that are further away than thisdistance. For example, the fluid dynamics model may be based onHele-Shaw flow, which may provide a more global solution. In Hele-Shawflow, the velocity flow is solenoidal, which useful to obtain thevelocity of the force point due to its own force. The fundamentalsolution for (depth averaged) Hele-Shaw flow is simply a replacement forthe Stokeslet. Thus, the velocity response due to a point force for theHele-Shaw flow may be determined by replacing the function G (e.g., ofequation 2) related to 2D Stokes flow with a different function Grelated to Hele-Shaw flow.

In an embodiment, referring to FIGS. 8A and 8B, there is provided amethod 800 for determining resist deformation and further apply theresist deformation model to adjust parameters of the patterning process.In an embodiment, the resist deformation model may be obtained andsimulated to determine the deformation.

The method, in process P82, involves obtaining a resist deformationmodel 801 of resist having a pattern. In an embodiment, the resistdeformation model is configured to simulate a fluid flow of the resistdue to capillary forces acting on a feature contour of the pattern.Further, in an embodiment, an input pattern 803 (e.g., as in FIG. 3C)may be obtained to be processed by the resist deformation model 801. Inan embodiment, some other quantity that represents the presence of afeature at that location is thresholded to obtain the featureboundaries. Examples of other quantities include but not limited to ared component of a color image, a blue component of a color image, agreen component of a color image, the hue of an image, the saturation ofan image, pixel values of an image, etc. The input pattern 803 may alsobe the output from a convolutional filter or another edge detectionfilter.

In an embodiment, the resist deformation model may be obtained asdiscussed for example, as discussed above with respect to FIGS. 4A and5. In an embodiment, the resist deformation model is a fluid dynamicsmodel. In an embodiment, obtaining the resist deformation model involvesgenerating the model including defining a fluid dynamic model anddefining boundary conditions, as discussed in FIGS. 4A and 5. In anembodiment, obtaining the resist model involves receiving the resistmodel via a network. In an embodiment, the resist model may be receivedfrom a database or the process may be configured to communicate withanother process on which the resist model is implemented. In anembodiment, the fluid dynamics model is based on a linearizedNavier-Stokes flow equations, as discussed in FIG. 4A and FIG. 5. Forexample, the fluid flow is characterized by a 2D Stokes flow (an exampleof Stokes flow) or a Hele-Shaw flow.

The method, in process P84, involves determining, via a processor (e.g.,processor 104) and simulation of the resist deformation model (e.g., 2DStokes flow based equations 2-11 and corresponding boundary conditionsin FIGS. 4E-4H), a deformation of a developed resist pattern for aninput pattern 803 to the resist deformation model. Depending on theboundary conditions and the velocity field, the deformations obtainedmay vary. For example, as discussed with respect to FIG. 4G showsdeformation corresponding to wall boundary condition, 4H showsdeformation when periodic boundary condition is applied, FIG. 7illustrates a deformation of a part of a resist domain.

In an embodiment, the simulation further involves following processessuch as P844-P848 discussed with respect to FIG. 8B. In an embodiment,the simulation involves defining initial vertices along the featurecontour, in process P842. In an embodiment, the initial vertices may berearranged to satisfy a particular condition such to stabilize thesimulation process. For example, process P844 involves redistributingthe vertices to make them evenly spaced while conserving an area or avolume of the feature contour of the pattern.

Further, the simulation process, in process P846, involves determining acapillary force at a given vertex along the feature contour of thepattern. In an embodiment, the capillary force acting on the givenvertex is a sum of tensions on either side of the given vertex. Theforce can be determined, for example, using equations 3-5. In anembodiment, the forces may cause a squeeze flow which is a flow ofresist due to a net inward flux or a net outward flux through thevertical resist domain boundaries causing a large scale migration of thefeatures. Such squeeze flow (e.g., as shown FIG. 6) may be eliminated byapplying appropriate boundary conditions as discussed with respect toFIG. 4A.

Process P848 involves applying a boundary condition to the fluiddynamics model, for example, to eliminate a squeeze flow. Examples ofboundary conditions and how to implement such boundary conditions arediscussed with respect to FIGS. 4A-4H, earlier in the disclosure.

Once the forces and boundary conditions are applied, process P850involves determining a velocity field 850 (an example of 520 of FIG. 5and 721 of FIG. 7) of the fluid flow due to the capillary forces basedon superposition of Stokeslets and the boundary conditions. In anembodiment, the process P850 involves obtaining velocity at a givenvertex along the feature contour based on the velocities of all othervertices due to a capillary force at the given vertex. This is achievedby multiplication of the force at the given vertex with the Stokesletcentered on the given vertex, an evaluating it at the other vertices.The velocity at the given vertex due to the force at the first vertex issuch that the velocity conserves the area or volume of the feature. Inan embodiment, the velocity is in the normal direction to the force.

In an embodiment, the squeeze flow may be determined by decomposing thevelocity field and determining appropriate boundary conditions toelimination such squeeze flow. For example, process P852 involvesdecomposing the velocity field 850 in to the squeeze flow (e.g., 600 ofFIG. 6) and a higher order velocity flow (not illustrated). Then,process P854 involves eliminating the squeeze flow from the velocityfield by applying the boundary condition. In an embodiment, the boundarycondition comprises: setting a flow rate through the boundary of theresist to zero; and/or setting a velocity across the boundary of theresist to no-flux condition. As discussed earlier, the flow rate intothe domain through the vertical boundaries is set to zero by providing acombination of rotlets (e.g., see FIG. 4E) of appropriate strengths(e.g., using equation 10 and as discussed with respect to FIG. 4E) atthe corners of the boundary of the resist domain. In an embodiment, therotlets of equal magnitude and alternating signs at are placed at fourcorners of the resist domain.

Turning back to FIG. 8A, the method, in process P86, may optionallyinvolves determining forces at a partial area within the resist andobtaining a deformation of the entire area of the resist based on thesimulation of the resist deformation model using forces at the partialarea, for example, as discussed with respect to FIG. 7.

In an embodiment, the method may involve, in process P88, simulation forobtaining the resist deformation at a desired instance of time viasimulation of the resist deformation model till the desired instance ofthe time. In an embodiment, contributions to the velocity field due tofeatures further away from a region where the capillary forces areapplied are negligible.

As discussed earlier, the input pattern (e.g., FIG. 3C) is provided toresist deformation model in the form of an image of the input pattern.The image may be a binary image. In an embodiment, the input pattern isa design pattern, a resist image, a mask pattern, and/or an aerialimage. In an embodiment, obtaining the input pattern comprisesgenerating the binary image. The binary image may be generated byobtaining a patterning device pattern corresponding to the inputpattern, producing, via simulation of patterning process, an aerialimage based on the patterning device pattern; and extracting boundariesof the pattern in the aerial image to generate the binary image.

In an embodiment, the above methods 400, 500 or 800 may further involvecomputing, using the resist deformation model, a critical dimensionbetween a pair of locations disposed on a boundary of the developedresist pattern and calculating an error between the computed criticaldimension and a measured critical dimension of an actual developedresist pattern. Such CD and error values may be further used to performoptimization of the patterning process, for example, OPC, maskoptimization, source optimization or a combination thereof.

In an embodiment, the deformation is determined at a plurality oflocations, each location corresponding to a point that lies on aboundary of a developed portion of the developed resist pattern for theinput pattern. In an embodiment, the resist is a negative tone resist ora positive tone resist, chemically or not-chemically amplified.

FIG. 9 is a flow chart of a method 900 for determining a parameter of apatterning process based on a resist deformation computed based ondeformation model discussed above. The method 900, in process P92,involves obtaining (i) a patterning process model 901 that includes aresist deformation model (e.g., discussed with respect to methods FIG.4A and FIG. 5) of a resist having a pattern, and (ii) a target pattern903 (e.g., a design pattern). The resist deformation model configured tosimulate a fluid flow of the resist due to capillary forces acting on acontour of at least one feature of the pattern. In an embodiment, theresist deformation model is a fluid dynamics model configured tosimulate a fluid flow of the resist due to capillary forces acting on afeature contour of the pattern.

Process P94 involves determining, via a processor (e.g., processor 104),a resist pattern 904 based on a simulation of the patterning processmodel 901 with the target pattern 903 as an input to the patterningprocess model, where a difference exists between the resist pattern andthe target pattern. Examples of resist patterns within the resistdeformation (e.g., 520) of an example input pattern (e.g., 500 in FIG.5) are illustrated in FIGS. 5 and 7. The deformation process based onthe forces and velocity based on 2D Stokes flow is discussed withrespect to the method 400 of FIG. 4A.

Process P96 involves determining, via the processor, a value of aparameter of the patterning process based on the simulation of thepatterning process, the value of the parameter being determined suchthat the difference between the resist pattern and the target pattern isreduced. In an embodiment, the parameter of the patterning processcomprises at least one of dose, focus, optical proximity correction. Forexample, the resist deformation model may be included in the resistmodel of the resist process of the patterning process. Simulation of thepatterning process using such resist model can allow optimization of thepatterning process including determining optimum values of parameterssuch as dose, focus, optical parameters, OPC, etc. The optimizationprocess may involve reducing a cost function including the differencebetween the resist pattern and the input pattern.

Process P98 involves applying the value of the parameter of thepatterning process to a lithographic apparatus during the patterningprocess. A wafer printed based on such parameter values may be furthermeasured (e.g., using optical tools or SEM) and used to verify theresults of the deformation model. For example, a SEM image of theprinted wafer obtained via a metrology tool. The SEM image may befurther processed to determine resist patterns which can be compared(e.g., EPE between patterns of simulation image and SEM image) with thedeformation obtained at a desired instance of the simulation of thedeformation model. Based on the comparison, the fluid dynamics model maybe validated or modified. For example, additional error or penalty termsmay be included in the deformation model. The error or penalty terms mayaccount for such difference in simulated and actual measured values ofthe substrate.

While embodiments have been described in terms of binary images as theresist pattern image, in an embodiment, the input resist pattern imagecan be grayscale and/or the output deformed resist pattern image can begrayscale. Further, while embodiments have been described in terms ofusage of images, it will be appreciated that the resist pattern can bemore generally characterized in terms of data, such as CDs, coordinatelocations, vectors, etc. and so the input resist pattern data and/or theoutput deformed resist pattern resist deformation data can be in anon-image form such as CD values, coordinate locations, vectors, etc.

In an embodiment, the fluid dynamics model can be relatively fast byadopting the binarization of the resist pattern image data. Additionallyor alternatively, simplification of the Navier-Stokes equations forfaster application towards full-chip solutions can be done bydiscretizing the equations and representing them by a sum of kernelfunctions.

So, in sum, the effect of intrafluid forces, such as surface tension, ona resist pattern is included in a prediction of a resist pattern using acomputational fluid dynamics model. Data regarding the resist pattern(such as an optical image (or image derived therefrom) produced using,e.g., ASML's Tachyon product) is used an input to the model. In anembodiment, to speed processing, it is binarized to areas that areassumed to be fully developed and a remainder. The remainder resist istreated as a fluid on a laminar two-phase flow field and with constantor non-constant viscosity. Then, intrafluid forces are effectivelymodeled such as surface tension applied on the boundaries of thedeveloped resist. The model determines, for example, fluid velocity andpressure and consequently the deformation of the resist are calculatedby solving fluid dynamics equations such as the Navier-Stokes equations.Thus, fluid dynamics are used to enable resist profile prediction andthe fluid dynamics model can efficiently include, e.g., the surfacetension effects. In particular, in an embodiment, strain and curvatureeffects of highly dense patterned shapes are captured by using fluiddynamics.

To make the model suitable for a particular patterning process wherevalues of physical and materials parameters are not known (e.g., wherethe viscosity, density, etc. is not known), the model can be fitted(e.g., by regression) to measured values on actual deformed features(e.g., experimental CD values in X and Y directions).

Accordingly, in an embodiment, there is provided a method thatdetermines resist deformation which balances accuracy and speed to allowrelatively easy integration of resist deformation prediction intoexisting algorithms for patterning process configuration. For example,in an embodiment, there is provided a fluid dynamics model than canprovide better prediction of the deformation of a resist pattern thanpast modeling. Additionally or alternatively, the fluid dynamics modelcan provide faster deformation predictions than past modeling.

During the development stage, a solvent diffuses into the resist 1010 ora feature therein, which softens an outer layer (e.g., 1020) of theresist, as shown in FIGS. 10A and 10B. One way of modelling suchbehavior is based on assumption that (i) capillary and viscous floweffects are dominant, and (ii) continuum description within the resistand/or a liquid layer associated with the resist holds.

Capillary forces deform this outer layer 1020, either elastically or viaviscous flow, until the solvent has evaporated again. Since bothdeformation processes are approximately governed by the similarequations for small deformations, the flow of a thin layer is consideredas viscous flow. According to an embodiment, a thin liquid layer 1020(also referred as a boundary liquid layer 1020) of the resist 1010around a feature boundary will flow, driven by capillary forces. As thefeature boundary and a liquid layer around the feature are associatedwith each other and any deformation in the liquid layer causes a changein shape of the feature, the terms “boundary liquid layer,” “thin-film,”and “contour,” or “boundary,” of a feature, may be used interchangeablyherein. Accordingly, a deformation of the boundary liquid layer refersto a deformation of a boundary/contour of a feature. In an embodiment,features may represented by their boundaries (e.g., in FIG. 10A) at aspecified height H with respect to a surface of the substrate (e.g., asshown in FIG. 10B).

In an embodiment, the fluid flow of the boundary liquid layer 1020 isdecomposed into two components: a horizontal flow and a vertical flow,as shown in FIGS. 10A and 10B, respectively. Accordingly, a deformationof the boundary liquid layer (and/or the feature associated therewith)caused by the fluid flow may be a resultant of a first deformationcomponent caused by the horizontal flow component (e.g., 1011, 1012,1013, 1015, and 1016) and a second deformation component caused by thevertical flow component (e.g., 1021, 1022, 1023). In an embodiment, theboundary of a feature may have a net inward motion at a given height(e.g., H) above the substrate.

As shown in FIG. 10A, the horizontal flow component (e.g., 1011, 1012,1013, 1015, and 1016) in the horizontal plane acts along a boundary ofthe layout or geometric shape of the feature (e.g., a line or a circle).The horizontal flow component causes a change this shape of the boundaryliquid layer 1020, but not the total area of the feature. In anembodiment, the horizontal flow depends on a curvature of the feature inthe horizontal plane.

As shown in FIG. 10B, the vertical flow component in a vertical planeacts along the shape of a cross section in the vertical plane of theboundary liquid layer (or the feature associated therewith). Thevertical flow can change the total area. The vertical flow depends onthe curvature in the vertical direction

Thus, a resist deformation model (e.g., a thin-film model) is configuredto determine the first deformation caused due to the horizontal flowcomponent of the fluid flow and further adjust the first deformation toaccount for a second deformation caused by the vertical flow component.

In an embodiment, the resist deformation model is configured to capturedifferent aspects (e.g., flow rates, deformations, conservation of massand/or volume, etc.) of the horizontal flow and the vertical flowcomponents. Furthermore, according to a thin film approximation, thedeformation of any feature is independent of the deformation of theother adjacent features. This causes the computation time to be linearin the number of features, which is the best possible scaling. Finally,a direct implementation of the model parallelizes very well, since anypoint (along a contour of the feature) only interacts with its nearestneighbors and its next-nearest neighbors. Furthermore, the resistdeformation model is configured to determine deformation of thefeatures, due to shrinkage based on the lubrication approximation. Inthe lubrication approximation, a vertical shear stress is neglected. Thevertical shear stress is negligible if the deformation changes onlygradually in the horizontal direction, over distances much longer thanthe layer thickness.

FIG. 10C depicts a schematic top view of boundary liquid layer adjacentto the feature in the resist according to the present disclosure. FIG.10C is a top view of a model of an exposed resist having the boundaryliquid layer (i.e., a gel-type layer) defined. In this model, a boundaryliquid layer 930 having a width 950 (also referred as a layer thicknessδ) is defined for the developed or open regions 910 (e.g., trenches).The developed or open region 910, such as a trench, between portions ofthe resist 900 is formed by the development. In this example, the region910 would have gas therein.

The resist 900 is specified so as not to deform at all or very little bybeing specified as a solid with a relatively high modulus of elasticityor a liquid with a high viscosity compared to the boundary liquid layer930 (e.g., 30% or more greater, 50% or more greater, 75% or moregreater, 100% or more greater, 200% or more greater, 500% or moregreater, or 1000% or more greater). Indeed, the resist 900 may not evenneed to be specified in the model and instead a boundary condition isapplied to the boundary liquid layer 930 that is tantamount tospecifying an adjacent region of high viscosity or an adjacent regionincapable of any or much deformation. The boundary liquid layer 930 thusexhibits all, or most, of the deformation into the region 910. In thisexample, the extension of the boundary liquid layer 930 on top of theresist 900 is not shown. However, all, or part of, the resist 900 may becovered with boundary liquid layer 930.

The resist deformation model is characterized in terms of a boundaryliquid layer 930 located at a boundary between a developed or openregion of the resist pattern and the resist 900. The boundary liquidlayer 930 has a width smaller than the width of the resist at theboundary, the parameter of, or associated with, the boundary liquidlayer can be varied subject to an appropriate boundary condition at theboundary of the feature (at the open region) and at a side of theboundary liquid layer opposite to the developed or open region. Forexample, the boundary condition applied to the boundary liquid layer canbe tantamount to specifying an adjacent region of high viscosity or anadjacent region incapable of any or much deformation.

In an embodiment, a resist deformation model is developed under theassumption that capillary and viscous flow effects are dominant.Furthermore, the resist deformation model is developed under theassumption that continuum description holds. Hence, applicability of themodel for physical results is limited to a length scale and/or a timescale (e.g., width, a time period of simulating the deformation, etc.).In an embodiment, the width may be a pre-determined amount, e.g.,specified by a user and desirably selected such that it is larger thanthe largest expected deformation toward the essentially insolubleresist. In an embodiment, the width is selected from a range of from 5nm to 300 nm, a range of from 5 nm to 200 nm, a range of from 5 nm to100 nm, a range of from 5 nm to 50 nm, a range of from 10 nm to 40 nm,or a range of from 5 nm to 30 nm, or a range of from 5 nm to 20 nm. Insuch an embodiment, the width may not be varied as part of thecalibration routine. Or, in an embodiment, the pre-determined width maybe a starting point and the width may be varied like a parameter of, orassociated with, viscosity, etc.

In an embodiment, the width may be a range (such as the ranges describedjust above), which range can act as a constraint. Thus, in such anembodiment, a certain width can be a starting point and the width may bevaried like a parameter of, or associated with, viscosity, etc., butconstrained within the range given.

In an embodiment, the width can vary at different locations along theboundary liquid layer. For example, the width can be different indifferent developed or open regions or at different parts along adeveloped or open region. As another example, the width can be differentat an upper surface (e.g., in top view) of the resist rather than in asidewall (e.g., in side view) of a developed region. In an embodiment, arelationship (e.g., a ratio) can be specified between the sidewall widthand the upper surface width so as to constrain the difference in width.

So, in an embodiment, when short-range interaction of forces isdesirable, a region of liquid material having a finite (and relativelysmall such as 30 nm or less) width can be defined at a boundary of adeveloped or open region and a resist, the region at the boundary havinga width smaller than the resist at the boundary. On the side opposite ofthe developed or open region, the model can have another material, e.g.,a solid, that does not deform or deforms significantly less than thematerial of the region having the finite width (e.g., has asignificantly higher viscosity than the material of the region havingthe finite width) or have a boundary condition that tantamount specifieslow or no deformation at that location. As a result, the velocity fieldat the opposite side can be at or close to zero, thus leading to littleor no changes in deformation at that location (and at outward locationstherefrom until another region of finite width is encountered).

FIG. 11 depicts an exemplary schematic output obtained by simulation ofthe resist deformation model using, for example, pattern information(e.g., contours extracted from an aerial image, as discussed earlierwith respect to FIG. 3C). Specifically, FIG. 11 depicts (i) initialcontours Mi (e.g., an input contour Mi1, Mi2, and Mi3 extracted fromAI), (ii) deformed contours M1 o (comprising contours 111, 1112, and1113) obtained from a full-fluid dynamics model (i.e., where the entireresist is treated as liquid instead of a thin layer of resist), and(iii) another deformed contours M2 o (comprising contours 1121, 1122,and 1123) obtained from the resist deformation model according topresent disclosure.

In FIG. 11, the deformed contours 1111 and 1113 (which are part ofoutput M1 o) obtained from the full-fluid model predicts (e.g., M1 o)that the contours are deformed outward into the resist 1010 relative tocorresponding input contours Mi1 and Mi2, respectively, and the deformedcontour 1112 is deformed inward relative to the corresponding inputcontour Mi3. Such deformed contours 1111, 1112 and 1113 are caused dueto interaction between deformation of the individual contours Mi1, Mi2,and Mi3, since the entire resist 1010 is assumed to be fluid. In otherwords, deformation of one feature (or the contour thereof) may cause adeformation of a neighboring feature (or the contour thereof).

The deformed contours 1121 and 1123 (which are part of output M2 o)obtained from the thin film fluid model (of present disclosure) predicts(e.g., M2 o) that the contours are deformed inward relative tocorresponding input contours Mi1 and Mi2, respectively, and the deformedcontour 1122 is not deformed or slightly deformed relative to thecorresponding input contour Mi3. Within such model, the deformed of onefeature is independent of the other neighboring feature, since only athin layer around the feature is assumed to deform due to fluid flowwhile regions of resist 1010 further from the thin layer do not deformor only negligibly deform. As can be seen, there is relatively slight,but significant, deformation at curved portions of the contours such as1121 with respect to the input contour Mi1, and is much different fromthe deformation of contour 1111 at similar curved portions of the inputcontour Mi1.

In an embodiment, the resist deformation model comprises formulasrelated to a thin-film model such as (i) a momentum equation simplifiedaccording to the lubrication approximation, (ii) conservation of massand volume, and (iii) boundary conditions. The following descriptionprovides example formulation of the resist deformation model todetermine a deformation due to the horizontal flow component and thevertical flow component of the fluid flow of the boundary liquid layer.

In an embodiment, the horizontal flow component of the fluid flow of theboundary layer (e.g., 930) can be determined based on a pressure drivenlubrication flow, assuming that the viscosity of the boundary liquidlayer is constant. The horizontal flow is determined while conservingvolume of the features and volume of the boundary liquid layer. In thehorizontal flow, curvature of the contour of feature imposes limitingconditions on the way the deformation occurs. In an embodiment, thedeformation computations are performed on a plurality of vertices alonga contour of a feature, each vertex being a location on the contour ofthe feature. The computation also involves determining a Jacobian oftime derivative of deformation based on a finite difference method.

Further, as the concave portion of the contour moves the plurality ofvertices may be coincide, which is not desired. As such, the pluralityof the vertices a redistributed to adjust the spacing between verticesis performed.

In an embodiment, a pressure at a vertex may be determined usingfollowing equation.

$\begin{matrix}{P_{i} = \frac{{\overset{\rightarrow}{n}}_{i} \cdot {\overset{\rightarrow}{F}}_{i}}{r_{ci}}} & (12)\end{matrix}$

In the above equation, i is the i-th vertex of the plurality of vertex;n_(i) that is normal to a line between the nearest vertices of i-thvertex; r_(ci) mean distance to the nearest neighbors of i-th vertex,where if x is a location of the vertex then r_(ci) is given by ½|{rightarrow over (x)}_(i+1)−{right arrow over (x)}_(i−1)|; and {right arrowover (F)}_(i) is force at i-th vertex. In an embodiment, the force candetermined using, e.g., the Stokeslet based model, not using thefull-fluid model, but only the process to determine the force, discussedearlier.

According to the present disclosure, the fluid flow rate depends on theviscosity the boundary liquid layer thickness δ, and the pressuregradient ∂_(x)P in a horizontal plane.

To calculate the flow rate, velocity u is integrated over the layerthickness δ. According to the present model, the viscosity does notdepend on the shear rate of strain i.e., the boundary liquid isNewtonian. The viscosity may depend height H of the resist. Forsimplicity of explaining the concepts, the viscosity is assumed constantover the layer thickness. However, viscosity may vary.

The velocity can be determined based on a momentum equation in thelubrication approximation, and boundary conditions as described byfollowing set of equation:

∂_(y) ² u=∂ _(x) P  (13)

u=0 at y=0  (14)

∂_(y) u=0 at y=δ  (15)

In the above set of equation, y is the coordinate in the normaldirection, i.e., the distance from the solid surface (i.e., resist)around the feature, and x is a tangential coordinate. In an embodiment,a dynamic boundary condition set as the velocity gradient ∂_(y)u beingzero at a free surface (e.g., an outer edge representing a surface ofthe boundary layer at a counter of a feature). Further, a kinematicboundary condition is set as the velocity u is zero at the solid (e.g.,an inner edge representing a surface of the boundary layer at sideopposite to the contour of the feature).

As the pressure changes, the above equation may cause the layerthickness to take zero or negative values, which may make the simulationunstable. Hence, the layer thickness is determined midway between pointsi-th vertex and i+1-th vertex, as follows: δ=min(δ_(i),δ_(i+1))

Further, the pressure gradient ∂_(x)P is between points i-th vertex andi+1-th vertex (e.g., midway).

Then, the velocity can be computed using following equation:

u=(∂_(x) P)½(δ²−(δ−y)²)  (16)

Further, integrating of the velocity gives the flow rate Q (e.g.,equation below) of the boundary layer in the horizontal plane. Thisformula gives the flow rate Q from one vertex to the next vertex, as afunction of the pressure gradient and the layer thickness between thosevertices.

Q=∫ ₀ ^(δ) udy=(∂_(x) P)⅓δ³  (17)

Furthermore, a time derivative of a state variable comprising a layerthickness is determined per time step. For example,d_(t)δ=(Q_(i+1/2)−Q_(i−1/2))/r_(ci). In an embodiment, the timederivative of the thickness may be integrated to determine thickness atany given point.

The time derivative computation involves computation of Jacobian of timederivative, for efficient computation. The time derivative of a point(e.g., i-th vertex) depends on a position of its nearest neighbors andits next-nearest neighbors. Such Jacobian is a pentadiagonal matrix, andthus very sparse. The Jacobian matrix can be calculated in manydifferent ways. In an embodiment, the penta-diagonal refers to a maindiagonal (i.e., longest diagonal of a matrix such as passing through(1,1) to (n,n) in case of a square matrix) having non-zero elements andtwo diagonals adjacent to the main diagonal having non-zero elements,while remaining elements of the Jacobian matrix is zero. At firstglance, automatic differentiation (e.g., via pre-defined ODE solver) maybe employed. However, this may make the code harder to read and lessportable. For these reasons, automatic differentiation is undesirable.In an another embodiment, the Jacobian matrix may be computedanalytically. However, analytical computation require configuring andreconfiguring the governing equation and their derivative, which ishighly time consuming and prone to errors. On the other hand, accordingto the present model, the resulting Jacobian is sparse, which enablesuse of finite differencing may be used. The finite difference methodprovides faster computation time and less cost, as it takes only, forexample, 6 evaluations of the derivative when a vertex and it fourneighboring are used in computing the deformation.

Further consideration in determining the deformation due to horizontalflow involves concave corner or rounded edges. In concave corners,vertices will approach each other. Eventually, they will meet. When thathappens, the thin film approximation may become invalid, because aradius of curvature should be large with respect to the layer thickness.When the vertices meet, the radius of curvature is zero. Thus, whenvertices coincide, the normal direction is undefined, resulting in anunstable simulation. To mitigate this issue, the plurality of verticesare redistributed continuously, so that they cannot coincide. Thepresent disclosure is not limited redistributing vertices continuouslyi.e., at each time step, a person skilled in the art may performredistribution at discrete time interval as well.

In an embodiment, the redistribution of vertices to may be defined on atime scale that is at least of the order of magnitude as a time scale ofthe evolution of the layer thickness. By continuity, a decrease in layerthickness per unit time can be determined as the derivative of the flowrate with respect to the tangential coordinate. The time scale soobtained may be different for different vertices.

Further, it is determined how to redistribute the vertices, and how thatchanges the layer thickness of each vertex. An example redistribution isbased on three specifications as follows. First, the redistributionshould conserve the total volume of the feature. This is achieved bymoving any vertex only in a direction parallel to the line between thenearest neighbor vertices. Second, the redistribution should converge,i.e., it should not have any growing modes. This can be achieved byusing diffusion of a vertex spacing as the redistribution, where a speedof each vertex is proportional to its distance to a line halfway betweenthe nearest neighbor vertices. The constant of proportionalitydetermines the time scale of the redistribution. Third, the volume ofthe boundary liquid layer should be conserved. This can be achieved byevaluating the geometric representation of the boundary liquid layer.

Referring to FIG. 12A, the boundary liquid layer is represented by alayer thickness and vertex positions. An inner edge BLi, between theboundary liquid layer and the solid part (i.e., resist), is drawn with adashed line. An outer edge BLo is drawn with a solid line and dots 1208,1209, 1210, 1211 and 1212 are sample vertices. For the middle vertex1210 (e.g., an i-th vertex), a centerline, a tangent line 1230, and adirection of motion are shown. Of each vertex on the outer edge (i.e.,at the contour of the feature), a position and a layer thickness are thestate variables. With redistributions, the normal and tangential vectorsare obtained from the instantaneous positions of the outer edge BLovertices 1208, 1209, 1210, 1211 and 1212.

The redistribution moves outer vertices (i.e., vertices on the outeredge BLo of the boundary layer) in the direction of the line between theneighboring outer edge vertices. If the normal directions would notchange, inner edge vertices (i.e., vertices along the inner edge BLi)(not shown) would move in the direction of the line between theneighboring inner edge vertices. An additional change in layer thicknessmust by imposed to take the change in normal direction into account.When one outer vertex moves in the tangential direction, this causes thecorresponding inner vertex and its nearest neighbors to move in thetangential direction of the corresponding outer vertices. The changes inlayer thicknesses must be such that their direction of motion is in thetangential direction of these inner vertices. In an embodiment, theredistribution is achieved based on the following time scale equationbelow, wherein Δx is a distance between consecutive vertices and λ isspatial period of 2Δx.

$\begin{matrix}{{\tau\left( {\lambda,\delta} \right)} = {- \frac{\pi^{4}\delta^{3}}{3\Delta\; x^{4}}}} & (18)\end{matrix}$

In an embodiment, the velocity in the inner normal direction should bezero to conserve the volume of the boundary liquid layer which resultsin the following equation:

$\begin{matrix}{{{\mathcal{u}} \cdot {\overset{\rightarrow}{e}}_{y,i}} = {{{{\mathcal{u}}_{x,i}\left( {{\overset{\rightarrow}{e}}_{x,o} \cdot {\overset{\rightarrow}{e}}_{y,i}} \right)} - {\partial_{t}{\delta\left( {{\overset{\rightarrow}{e}}_{y,o} \cdot {\overset{\rightarrow}{e}}_{y,i}} \right)}}} = {\left. 0\Rightarrow{\partial_{t}\delta} \right. = {{\mathcal{u}}_{x,i}\frac{{\overset{\rightarrow}{e}}_{x,o} \cdot {\overset{\rightarrow}{e}}_{y,i}}{{\overset{\rightarrow}{e}}_{y,o} \cdot {\overset{\rightarrow}{e}}_{y,i}}}}}} & (19)\end{matrix}$

In the above equation,

_(x,i) is the outer tangential velocity of the inner vertex; {rightarrow over (e)}_(x,o) refers to outer tangential direction; {right arrowover (e)}_(y,i) refers to inner normal direction; and {right arrow over(e)}_(y,o) refers to outer normal direction. The above equation 19,gives the time rate of change of the layer thickness that conserves thevolume of the boundary liquid layer, under redistribution of vertices.This time rate of change of the layer thickness conserves the volumewhen the outer vertices move in the outer tangential direction. Itshould be added to the time rate of change of the layer thickness due toflow, which causes the outer vertices to move in the outer normaldirection.

Furthermore, when a neighboring outer vertex moves, this changes thenormal direction. The time derivative of the normal vector depends onthe velocity of the neighboring vertex and on the distance between theneighboring vertices. This is because the neighboring vertices determinethe tangential direction. Now that the normal vectors can change duringthe simulation, the normal vector should be accounted for in change dueto flow. The flow of material causes a component of the velocity of anouter vertex in the outer normal direction. This changes the normaldirection of the neighboring vertices, which changes the innertangential velocity.

The end of redistribution process results in the first displacementcomponent due to the horizontal flow of the boundary layer. Further, thefirst displacement component is adjusted to include the seconddisplacement due to the vertical flow of the fluid flow.

In an embodiment, the vertical flow component of the fluid flow of theboundary layer (e.g., 930) can be determined as follow. The calculationof the vertical flow is (for modelling purposes assumed to be)independent of the horizontal flow. Thus, in an embodiment, deformationdue to only horizontal flow, only due to vertical flow may, or acombination thereof may be determined. In an embodiment, the verticalflow component results in an additional component of the time derivativeof the layer thickness, which is used to determine deformation of theboundary layer. The deformation is calculated in the lubricationapproximation, where vertical shear stress is neglected. Vertical shearstress is negligible if the deformation changes only gradually in thehorizontal direction, over distances much longer than the layerthickness.

In an embodiment, the vertical flow is determined based on a resistshrinkage model. FIG. 12B illustrates example nomenclatures used todefine the vertical flow of the resist deformation model and theshrinkage model is defined as follows:

$\begin{matrix}{{\partial_{t}\delta} = {{\overset{.}{\delta}}_{h} + {\frac{{\kappa\pi}^{3}}{8h^{3}}\mspace{14mu}\sin\frac{\pi\; y}{2h}}}} & (20)\end{matrix}$

In the above equation and referring to FIG. 12B, y is the coordinate inthe normal direction, i.e., the distance from the solid surface of thefeature; x is the tangential coordinate; {dot over (δ)}_(h) is the timederivative of the layer thickness due to horizontal flow, δ is the layerthickness, κ is the volumetric shrinkage; and h is the height of thefeature.

To calculate the vertical flow, extract the shape of the sides (e.g., asshown in FIG. 10B) from the input pattern information and apply theresist deformation model due to shrinkage based on the lubricationapproximation. Further, it is assumed that the deformation isincompressible. In an embodiment, the deformation is 2-dimensionaldeformation, which would be a good approximation for the deformation of,for example, a line feature. However, the present formulation is nolimited to a particular shape of the feature, and can be extended forany feature shape.

The resist deformation model in vertical flow determination is based ona formulation where the divergence of the stress τ_(i,j) vanishes inmaterial of the feature, since no body forces act on the material. Allforces act on the boundaries. Accordingly, an elasticity formulationthat describes displacements in a boundary liquid layer, where stressesand strains are eliminated from formulation may be employed. Forexample, a linear elasticity formulation defined by Navier-Cauchyequation may be employed along with the above assumptions to determinehow the boundary liquid layer deforms due to the vertical flow.

In an embodiment, the Navier equation is used to obtain the governingequation in terms of a horizontal displacement (in x-direction) only,given as follows:

2∂_(x) ² u _(x)+∂_(y) ² u _(x)=0  (21)

Further to the above equation, in an embodiment, the boundary conditionat the top, y=δ, is that the stress should vanish, since there is nomaterial beyond the feature edge to pull on. And, the boundary conditionat the bottom, y=0, is that the displacement should vanish. An exampleof a deformed shape of the cross-section of the resist in vertical planeis shown in FIG. 13. In FIG. 13, the edges 1301, 1303 and 1305 are awayfrom the resist and deformed due to the vertical flow as discussedabove. It can be seen that the edge at the solid portion or the resistis not deformed, as per the boundary condition.

FIG. 14A is a flow chart of example method 1400 for determining, basedon a thin-film based fluid model simplified according to the lubricationapproximation, a deformation of a pattern to be formed in resist duringa patterning process. In an embodiment, a first deformation of thefeature is simulated based on the fluid flow in a first plane (e.g., ahorizontal flow). Further, based on a fluid flow in a second plane(e.g., vertical plane), the first deformation is further deformed(referred as a second deformation) to account for the vertical flow asdiscussed earlier. The result is a deformation of the pattern thataccounts for deformations due to both horizontal flow and the verticalflow.

The method 1400, in process P141, involves inputting, into a resistdeformation model, pattern information 1401 relating to the pattern tobe formed, the model configured to simulate deformation of a portion ofa resist, the portion comprising a boundary liquid layer (e.g., 930 inFIG. 10C) located at a boundary between a developed region in the resistand a region of the resist surrounding the developed region,

In an embodiment, pattern information 1401 is in form of an imagecomprising a contour(s) of a feature(s) as input. In an embodiment, thecontours are extracted (e.g., in 320) from an aerial image (e.g., in310) as discussed with respect to FIGS. 3B and 3C above. The extractedcontours can be a grey scale image or a binary image. In an embodiment,the extracted contour refers to points where the force due to surfacetension will be applied.

Process P143 involves determining the first deformation 1410 based onthe horizontal flow as discussed earlier. In an embodiment, thedetermining the deformation further includes defining a plurality ofvertices (e.g., in FIGS. 12A and 15) along a contour of the developed oropen region in the resist; determining a capillary force {right arrowover (F)}_(i) at a given vertex along the contour; and redistributing ofthe vertices such that (i) a volume of the developed region in theresist is conserved, (ii) the redistribution should converge, and (iii)a volume of the boundary liquid layer is conserved. An exampleredistribution process is described earlier as related to the horizontalflow and FIG. 12A.

For example, determining deformation due to horizontal flow involvesdetermining forces {right arrow over (F)}_(i) at each vertex of theboundary layer (the contour of the feature), a velocity and flow rate ateach vertex of the contour, and redistribution of the vertices toobtained the first displacement of the vertices on the boundary layer atthe developed region (i.e., the contour of the feature), of the verticesat an edge (at the resist) opposite to the developed region. FIG. 14B isan example flow chart of method for determining deformation 1410 basedon the horizontal flow.

In FIG. 14B, the process of determining the first deformation using thepattern information 1401 is an iterative process involving processesP1431, P1433, P1435, and P1437. Process P1431 involves determining force{right arrow over (F)}_(i) at a vertex of the plurality of verticesusing, e.g., equations 3-5 as discussed earlier. The force {right arrowover (F)}_(i) at the vertex is further used to compute pressure {rightarrow over (F)}_(i) at the vertex using, e.g., equation 12 as discussedabove. Further, process P1433 involves determining velocity and flowrate based on a pressure gradient and the layer thickness. From thepressure in equation 12, the pressure gradient at a vertex is determinedand is further used to determine the velocity as discussed with respectto equations 13-16. Based on the velocity, the flow rate from one vertexto another can be determined using equation 17, as discussed above.Process P1435 involves redistributing the plurality of vertices based ona time scale, e.g., as defined by equation 18, and the threespecification (i) conserve the total volume of the feature, (ii) theredistribution should converge, and (iii) the volume of the boundaryliquid layer should be conserved, as discussed above. The redistributioninvolves determining a time scale based on the horizontal component ofthe flow rate of the boundary liquid layer, and moving, based on thetime scale, the vertices.

Furthermore, process P1437 involves determining the first deformation1410 using, for example, the time derivative of the flow rate (e.g.,equation 17). In an embodiment, the process P143 also involvesdetermining a rate of change of thickness (e.g.,d_(t)δ=(Q_(i+1/2)−Q_(i−1/2))/r_(ci)) of the boundary liquid layer basedon the flow rate of the boundary layer; and adjusting, based on theredistributed vertices (e.g., 1521 in FIG. 15), another plurality ofvertices (e.g., 1522 in FIG. 15) along the free surface based on therate of change of thickness of the boundary liquid layer.

The deformation 1410 due to the horizontal flow is adjusted based on thevertical flow, in process P145. The process P145 involves determiningthe second deformation 1420 based on the vertical flow, e.g., employingequations 20 and 21, as discussed earlier. The computation of thevertical flow is independent of the horizontal flow. In an embodiment,the second deformation is determined using a shrinkage model comprisinga time derivative of the layer thickness (e.g., 1410) due to horizontalflow and a height of the feature with respect to substrate. Thegoverning equation such that the divergence of the stress is zero.Further, the lubrication approximation is employed, where vertical shearstress is neglected. The assumptions lead to a simplified version ofNavier equation (e.g., equation 21) based on which a velocity and thedeformation 1420 of the boundary liquid layer and the feature associatedtherewith may be determined.

FIG. 15 illustrates an example of deformation of an input contour 1501(e.g., extracted from an aerial image, as discussed earlier) based onthe resist deformation model (e.g., based on thin-layer with lubricationapproximation). In FIG. 15, the input contour 1501 is further surroundedby another contour 1502 at a width δ, thereby forming a boundary liquidlayer 1510 having the width δ. Thus, in regards to employing the resistdeformation model discussed above, the contour 1501 serves as a contourat the feature and the another contour 1502 serves a contour away fromthe feature (i.e., at the resist which is assumed as solid). Further, aplurality of vertices are defined on the contours 1501 and 1502 and thedeformation of the boundary layer is determined with respect to theplurality of vertices. In an embodiment, the plurality of vertices onthe contour 1501 moves according to the velocity and the fluid flowequations discussed above, and further based on the pressure gradientacross the boundary layer and the width of the boundary layer, themovement of another plurality of vertices of the another contour 1502 isdetermined.

In an embodiment, the deformation of the vertices along the contour 1501is determined based on the horizontal flow formulation, as discussedabove, for example in method 1400, particularly process P143. Applyingthe formulation as discussed in process P143 results in a firstdeformation 1521 and 1521 of the contours 1501 and 1502, respectively.Thus, the deformed boundary layer 1520 due to the horizontal flow isobtained.

Further, the deformation of the vertices on the deformed contour 1521and 1522 is determined based on the vertical flow formulation, forexample, as discussed in process P145 above. Applying the vertical flowformulation results in deformed contours 1531 and 1532, as shown. Thus,the final deformed boundary layer 1530 that accounts for deformation dueto the vertical flow is obtained.

In an embodiment, there is provided a method for determining adeformation of a resist in a patterning process. The method includesobtaining a resist deformation model of a resist having a pattern, theresist deformation model configured to simulate a fluid flow of theresist due to capillary forces acting on a contour of at least onefeature of the pattern; and determining, via a processor and the resistdeformation model, a deformation of a resist pattern to be developedbased on an input pattern to the resist deformation model.

In an embodiment, the resist deformation model is based on a linearizedNavier-Stokes flow equations.

In an embodiment, the fluid flow is characterized by a Stokes flow,and/or a Hele-Shaw flow.

In an embodiment, the fluid flow is characterized by a fundamentalsolution that decays with distance to a particular location in theresist at least as fast as 1/R, and that is either regular in the resistor only singular at the location in the resist.

In an embodiment, determining the deformation involves defining aplurality of vertices along the contour of the at least one feature;determining a capillary force at a given vertex along the contour of theat least one feature; and determining a velocity flow field of the fluidflow due to the capillary force based on a superposition of a velocityresponse at one or more of the plurality of the vertices of the verticesalong the contour of the at least one feature, and a boundary condition,where the velocity response is due to the capillary force acting at avertex adjacent to the given vertex.

In an embodiment, the defining the plurality of vertices includesdistributing the plurality of vertices to make them evenly spaced whileconserving an area or a volume of the contour of the at least onefeature.

In an embodiment, the capillary force acting on the given vertex is asum of tensions on either side of the given vertex. In an embodiment,the velocity response is characterized by Stokeslet, wherein a Stokesletis the velocity field due to a point force in a Stokes flow.

In an embodiment, the determining the velocity flow field furtherinvolves decomposing the velocity field into a squeeze flow and a higherorder velocity flow; applying a boundary condition corresponding to thesqueeze flow; and removing the squeeze flow from the velocity fieldbased on the boundary condition.

In an embodiment, the squeeze flow is a flow of resist due to a netinward flux or a net outward flux through the vertical domainboundaries, causing a large scale migration of the features.

In an embodiment, applying the boundary condition involves setting aflow rate through the boundary of a resist domain to zero; and/orsetting a velocity across the boundary of the resist domain to no-fluxcondition.

In an embodiment, the total inward flow rate through the vertical domainboundaries is set to zero by providing a rotlet of appropriate strengthat one or more corners of a boundary of the resist domain.

In an embodiment, the rotlets of equal magnitude and alternating signswhen traversing the domain boundary, at are placed at the four cornersof the rectangular resist domain. In an embodiment, determining avelocity flow field further involves obtaining velocity at the givenvertex along the contour of the at least one feature based on velocitiesof all other vertices caused due to the capillary force at each of theother vertices by multiplication with Stokeslets.

In an embodiment, contributions to the velocity flow field due tofeatures far away from the given vertex are negligible.

In an embodiment, the method further includes determining forces at apartial area within the resist; and obtaining a deformation of theentire area of the resist based on the simulation of the resistdeformation model using forces at the partial area.

In an embodiment, the input pattern is a grayscale image and/or a binaryimage. In an embodiment, the input pattern is a design pattern or anaerial image.

In an embodiment, the method further comprising generating the binaryimage, the generating involves obtaining a patterning device patterncorresponding to the input pattern; producing, via simulation of aprocess model, an aerial image based on the patterning device pattern;and extracting boundaries of the pattern in the aerial image to generatethe binary image.

In an embodiment, the method further involves computing, using theresist deformation model, a critical dimension between a pair oflocations disposed on a boundary of the at least one feature in thedeveloped resist pattern; and calculating an error between the computedcritical dimension and a measured critical dimension of an actualdeveloped resist pattern.

In an embodiment, the calculating the error comprises comparing, using across-correlation matrix between a printed wafer data and a simulatedimages.

In an embodiment, the deformation is determined at a plurality oflocations, each location corresponding to a point that lies on aboundary of a developed portion of the developed resist pattern for theinput pattern.

In an embodiment, the resist is a negative tone resist or a positivetone resist, chemically or not-chemically amplified.

In an embodiment, there is provided a method for determining a parameterof a patterning process, the method involves obtaining (i) a patterningprocess model that includes a resist deformation model of a resisthaving a pattern, the resist deformation model configured to simulate afluid flow of the resist due to capillary forces acting on a contour ofat least one feature of the pattern, and (ii) a target pattern;determining, via a processor, a resist pattern based on a simulation ofthe patterning process model with the target pattern as an input to thepatterning process model, wherein a difference exists between the resistpattern and the target pattern; and determining, via the processor, avalue of a parameter of the patterning process based on the simulationof the patterning process, the value of the parameter being determinedsuch that the difference between the resist pattern and the targetpattern is reduced.

In an embodiment, the parameter of the patterning process comprises atleast one of dose, focus, and optical proximity correction.

In an embodiment, the method further involves applying the value of theparameter of the patterning process to a lithographic apparatus duringthe patterning process.

In an embodiment, there is provided a method for determining adeformation of a pattern to be formed in a patterning process. Themethod involves inputting, into a resist deformation model, patterninformation relating to the pattern to be formed, the model configuredto simulate deformation of a portion of a resist, the portion comprisinga boundary liquid layer located at a boundary between a developed regionin the resist and a region of the resist surrounding the developedregion, where the model is configured to determine a first deformationcomponent of the boundary liquid layer caused by fluid flow of theboundary liquid layer and a second deformation component of the boundaryliquid layer caused by the fluid flow of the boundary liquid layer; anddetermining, via a processor, the deformation of the pattern to beformed in the resist based on the input pattern information, wherein thedeformation comprises a combination of the first deformation componentand the second deformation component of the boundary liquid layer.

In an embodiment, the boundary liquid layer has a thickness smaller thana length of the developed region in the resist at the boundary.

In an embodiment, the first deformation component is determined in ahorizontal plane based on a horizontal component of a flow rate of theboundary liquid layer and the second deformation is determined in thehorizontal plane based on a vertical component of the flow rate of theboundary liquid layer.

In an embodiment, the model has defined therein, a first boundarycondition comprising a velocity gradient approximately equal to zero ata free surface of the boundary liquid layer, the free surface beingopposite of the developed or open region.

In an embodiment, the model has defined therein, a second boundarycondition comprising a velocity approximately equal to zero at a surfaceat the developed or open region.

In an embodiment, the model has defined therein, the velocity as afunction of thickness of the boundary liquid layer and a pressuregradient across the thickness of the boundary liquid layer.

In an embodiment, the model has defined therein, the flow rate of theboundary liquid layer as a function of an integration of the velocityover the thickness of the boundary liquid layer.

In an embodiment, the model has defined therein, the deformation as afunction of the flow rate of the boundary liquid layer.

In an embodiment, determining of the deformation involves determiningthe first deformation component in a first plane caused by thehorizontal component of the flow rate of the boundary liquid layer;determining a final deformation in the first plane by adjusting thefirst deformation component based on the vertical component of the flowrate of the boundary liquid layer in a second plane, the second plane isperpendicular to the first plane.

In an embodiment, the first plane is the horizontal plane.

In an embodiment, the determining the deformation further involvesdefining a plurality of vertices along a contour of the developed oropen region in the resist; determining a capillary force at a givenvertex along the contour; and redistribution of the vertices such that(i) a volume of the developed region in the resist is conserved, and(ii) a volume of the boundary liquid layer conservation.

In an embodiment, the redistribution of the vertices involvesdetermining a time scale based on the horizontal component of the flowrate of the boundary liquid layer, and moving, based on the time scaleand the flow rate of the boundary layer, the plurality of vertices.

In an embodiment, the method further involves determining a rate ofchange of thickness of the boundary liquid layer based on the flow rateof the boundary layer; and adjusting, based on the redistributedvertices, another plurality of vertices along the free surface based onrate of change of thickness of the boundary liquid layer.

In an embodiment, the model is a thin-film model simplified based onlubrication approximation, wherein the lubrication approximationcomprises a vertical shear stress value of zero.

Furthermore, there is provided a method for determining a deformation ofa pattern to be formed in a patterning process. The method includesinputting, into a resist deformation model, pattern information relatingto the pattern to be formed, the model configured to simulatedeformation of a portion of a resist, the portion comprising a boundaryliquid layer located at a boundary between a developed region in theresist and a region of the resist surrounding the developed region,wherein the model is configured to determine a deformation of theboundary liquid layer caused by a horizontal fluid flow of the boundaryliquid layer; and determining (e.g., in the process P143), thedeformation of the pattern to be formed in the resist by simulating theresist deformation model based on the input pattern information. Theboundary liquid layer has a thickness smaller than a length of thedeveloped region in the resist at the boundary.

In an embodiment, there is provided a non-transitory computer programproduct comprising machine-readable instructions for causing a processorto cause performance of the aforementioned methods.

FIG. 16 is a block diagram that illustrates a computer system 100 whichperform one or more aspects of the methods and flows disclosed herein.Computer system 100 includes a bus 102 or other communication mechanismfor communicating information, and a processor 104 (or multipleprocessors 104 and 105) coupled with bus 102 for processing information.Computer system 100 also includes a main memory 106, such as a randomaccess memory (RAM) or other dynamic storage device, coupled to bus 102for storing information and instructions to be executed by processor104. Main memory 106 also may be used for storing temporary variables orother intermediate information during execution of instructions to beexecuted by processor 104. Computer system 100 further includes a readonly memory (ROM) 108 or other static storage device coupled to bus 102for storing static information and instructions for processor 104. Astorage device 110, such as a magnetic disk or optical disk, is providedand coupled to bus 102 for storing information and instructions.

Computer system 100 may be coupled via bus 102 to a display 112, such asa cathode ray tube (CRT) or flat panel or touch panel display fordisplaying information to a computer user. An input device 114,including alphanumeric and other keys, is coupled to bus 102 forcommunicating information and command selections to processor 104.Another type of user input device is cursor control 116, such as amouse, a trackball, or cursor direction keys for communicating directioninformation and command selections to processor 104 and for controllingcursor movement on display 112. This input device typically has twodegrees of freedom in two axes, a first axis (e.g., x) and a second axis(e.g., y), that allows the device to specify positions in a plane. Atouch panel (screen) display may also be used as an input device.

According to one embodiment, portions of a process described herein maybe performed by computer system 100 in response to processor 104executing one or more sequences of one or more instructions contained inmain memory 106. Such instructions may be read into main memory 106 fromanother computer-readable medium, such as storage device 110. Executionof the sequences of instructions contained in main memory 106 causesprocessor 104 to perform the process steps described herein. One or moreprocessors in a multi-processing arrangement may also be employed toexecute the sequences of instructions contained in main memory 106. Inan alternative embodiment, hard-wired circuitry may be used in place ofor in combination with software instructions. Thus, the descriptionherein is not limited to any specific combination of hardware circuitryand software.

The term “computer-readable medium” as used herein refers to any mediumthat participates in providing instructions to processor 104 forexecution. Such a medium may take many forms, including but not limitedto, non-volatile media, volatile media, and transmission media.Non-volatile media include, for example, optical or magnetic disks, suchas storage device 110. Volatile media include dynamic memory, such asmain memory 106. Transmission media include coaxial cables, copper wireand fiber optics, including the wires that comprise bus 102.Transmission media can also take the form of acoustic or light waves,such as those generated during radio frequency (RF) and infrared (IR)data communications. Common forms of computer-readable media include,for example, a floppy disk, a flexible disk, hard disk, magnetic tape,any other magnetic medium, a CD-ROM, DVD, any other optical medium,punch cards, paper tape, any other physical medium with patterns ofholes, a RAM, a PROM, and EPROM, a FLASH-EPROM, any other memory chip orcartridge, a carrier wave as described hereinafter, or any other mediumfrom which a computer can read.

Various forms of computer readable media may be involved in carrying oneor more sequences of one or more instructions to processor 104 forexecution. For example, the instructions may initially be borne on amagnetic disk of a remote computer. The remote computer can load theinstructions into its dynamic memory and send the instructions over atelephone line using a modem. A modem local to computer system 100 canreceive the data on the telephone line and use an infrared transmitterto convert the data to an infrared signal. An infrared detector coupledto bus 102 can receive the data carried in the infrared signal and placethe data on bus 102. Bus 102 carries the data to main memory 106, fromwhich processor 104 retrieves and executes the instructions. Theinstructions received by main memory 106 may optionally be stored onstorage device 110 either before or after execution by processor 104.

Computer system 100 also preferably includes a communication interface118 coupled to bus 102. Communication interface 118 provides a two-waydata communication coupling to a network link 120 that is connected to alocal network 122. For example, communication interface 118 may be anintegrated services digital network (ISDN) card or a modem to provide adata communication connection to a corresponding type of telephone line.As another example, communication interface 118 may be a local areanetwork (LAN) card to provide a data communication connection to acompatible LAN. Wireless links may also be implemented. In any suchimplementation, communication interface 118 sends and receiveselectrical, electromagnetic or optical signals that carry digital datastreams representing various types of information.

Network link 120 typically provides data communication through one ormore networks to other data devices. For example, network link 120 mayprovide a connection through local network 122 to a host computer 124 orto data equipment operated by an Internet Service Provider (ISP) 126.ISP 126 in turn provides data communication services through theworldwide packet data communication network, now commonly referred to asthe “Internet” 128. Local network 122 and Internet 128 both useelectrical, electromagnetic or optical signals that carry digital datastreams. The signals through the various networks and the signals onnetwork link 120 and through communication interface 118, which carrythe digital data to and from computer system 100, are exemplary forms ofcarrier waves transporting the information.

Computer system 100 can send messages and receive data, includingprogram code, through the network(s), network link 120, andcommunication interface 118. In the Internet example, a server 130 mighttransmit a requested code for an application program through Internet128, ISP 126, local network 122 and communication interface 118. Onesuch downloaded application may provide for a process as describedherein, for example. The received code may be executed by processor 104as it is received, and/or stored in storage device 110, or othernon-volatile storage for later execution. In this manner, computersystem 100 may obtain application code in the form of a carrier wave.

FIG. 17 schematically depicts an exemplary lithographic projectionapparatus for use with the methods described herein. The apparatuscomprises:

-   -   an illumination system IL, to condition a beam B of radiation.        In this particular case, the illumination system also comprises        a radiation source SO;    -   a first object table (e.g., mask table) MT provided with a        patterning device holder to hold a patterning device MA (e.g., a        reticle), and connected to a first positioner to accurately        position the patterning device with respect to item PS;    -   a second object table (substrate table) WT provided with a        substrate holder to hold a substrate W (e.g., a resist-coated        silicon wafer), and connected to a second positioner to        accurately position the substrate with respect to item PS;    -   a projection system (“lens”) PS (e.g., a refractive, catoptric        or catadioptric optical system) to image an irradiated portion        of the patterning device MA onto a target portion C (e.g.,        comprising one or more dies) of the substrate W.

As depicted herein, the apparatus is of a transmissive type (i.e., has atransmissive mask). However, in general, it may also be of a reflectivetype, for example (with a reflective mask). Alternatively, the apparatusmay employ another kind of patterning device as an alternative to theuse of a classic mask; examples include a programmable mirror array orLCD matrix.

The source SO (e.g., a mercury lamp or excimer laser) produces a beam ofradiation. This beam is fed into an illumination system (illuminator)IL, either directly or after having traversed conditioning means, suchas a beam expander Ex, for example. The illuminator IL may compriseadjusting means AD for setting the outer and/or inner radial extent(commonly referred to as σ-outer and σ-inner, respectively) of theintensity distribution in the beam. In addition, it will generallycomprise various other components, such as an integrator IN and acondenser CO. In this way, the beam B impinging on the patterning deviceMA has a desired uniformity and intensity distribution in itscross-section.

It should be noted with regard to FIG. 17 that the source SO may bewithin the housing of the lithographic projection apparatus (as is oftenthe case when the source SO is a mercury lamp, for example), but that itmay also be remote from the lithographic projection apparatus, theradiation beam that it produces being led into the apparatus (e.g., withthe aid of suitable directing mirrors); this latter scenario is oftenthe case when the source SO is an excimer laser (e.g., based on KrF, ArFor F₂ lasing).

The beam PB subsequently intercepts the patterning device MA, which isheld on a patterning device table MT. Having traversed the patterningdevice MA, the beam B passes through the lens PL, which focuses the beamB onto a target portion C of the substrate W. With the aid of the secondpositioning means (and interferometric measuring means IF), thesubstrate table WT can be moved accurately, e.g. so as to positiondifferent target portions C in the path of the beam PB. Similarly, thefirst positioning means can be used to accurately position thepatterning device MA with respect to the path of the beam B, e.g., aftermechanical retrieval of the patterning device MA from a patterningdevice library, or during a scan. In general, movement of the objecttables MT, WT will be realized with the aid of a long-stroke module(coarse positioning) and a short-stroke module (fine positioning), whichare not explicitly depicted in FIG. 17. However, in the case of a waferstepper (as opposed to a step-and-scan tool) the patterning device tableMT may just be connected to a short stroke actuator, or may be fixed.

The depicted tool can be used in two different modes:

-   -   In step mode, the patterning device table MT is kept essentially        stationary, and an entire patterning device image is projected        in one go (i.e., a single “flash”) onto a target portion C. The        substrate table WT is then shifted in the x and/or y directions        so that a different target portion C can be irradiated by the        beam PB;    -   In scan mode, essentially the same scenario applies, except that        a given target portion C is not exposed in a single “flash”.        Instead, the patterning device table MT is movable in a given        direction (the so-called “scan direction”, e.g., the y        direction) with a speed v, so that the projection beam B is        caused to scan over a patterning device image; concurrently, the        substrate table WT is simultaneously moved in the same or        opposite direction at a speed V=Mv, in which M is the        magnification of the lens PL (typically, M=¼ or ⅕). In this        manner, a relatively large target portion C can be exposed,        without having to compromise on resolution.

FIG. 18 schematically depicts another exemplary lithographic projectionapparatus 1000 that can be used for the methods described herein.

The lithographic projection apparatus 1000 includes:

-   -   a source collector module SO    -   an illumination system (illuminator) IL configured to condition        a radiation beam B (e.g. EUV radiation).    -   a support structure (e.g. a mask table) MT constructed to        support a patterning device (e.g. a mask or a reticle) MA and        connected to a first positioner PM configured to accurately        position the patterning device;    -   a substrate table (e.g. a wafer table) WT constructed to hold a        substrate (e.g. a resist coated wafer) W and connected to a        second positioner PW configured to accurately position the        substrate; and    -   a projection system (e.g. a reflective projection system) PS        configured to project a pattern imparted to the radiation beam B        by patterning device MA onto a target portion C (e.g. comprising        one or more dies) of the substrate W.

As here depicted, the apparatus 1000 is of a reflective type (e.g.employing a reflective mask). It is to be noted that because mostmaterials are absorptive within the EUV wavelength range, the mask mayhave multilayer reflectors comprising, for example, a multi-stack ofmolybdenum and silicon. In one example, the multi-stack reflector has a40 layer pairs of molybdenum and silicon where the thickness of eachlayer is a quarter wavelength. Even smaller wavelengths may be producedwith X-ray lithography. Since most material is absorptive at EUV andx-ray wavelengths, a thin piece of patterned absorbing material on thepatterning device topography (e.g., a TaN absorber on top of themulti-layer reflector) defines where features would print (positiveresist) or not print (negative resist).

Referring to FIG. 18, the illuminator IL receives an extreme ultraviolet radiation beam from the source collector module SO. Methods toproduce EUV radiation include, but are not necessarily limited to,converting a material into a plasma state that has at least one element,e.g., xenon, lithium or tin, with one or more emission lines in the EUVrange. In one such method, often termed laser produced plasma (“LPP”)the plasma can be produced by irradiating a fuel, such as a droplet,stream or cluster of material having the line-emitting element, with alaser beam. The source collector module SO may be part of an EUVradiation system including a laser, not shown in FIG. 18, for providingthe laser beam exciting the fuel. The resulting plasma emits outputradiation, e.g., EUV radiation, which is collected using a radiationcollector, disposed in the source collector module. The laser and thesource collector module may be separate entities, for example when a CO2laser is used to provide the laser beam for fuel excitation.

In such cases, the laser is not considered to form part of thelithographic apparatus and the radiation beam is passed from the laserto the source collector module with the aid of a beam delivery systemcomprising, for example, suitable directing mirrors and/or a beamexpander. In other cases the source may be an integral part of thesource collector module, for example when the source is a dischargeproduced plasma EUV generator, often termed as a DPP source.

The illuminator IL may comprise an adjuster for adjusting the angularintensity distribution of the radiation beam. Generally, at least theouter and/or inner radial extent (commonly referred to as σ-outer andσ-inner, respectively) of the intensity distribution in a pupil plane ofthe illuminator can be adjusted. In addition, the illuminator IL maycomprise various other components, such as facetted field and pupilmirror devices. The illuminator may be used to condition the radiationbeam, to have a desired uniformity and intensity distribution in itscross section.

The radiation beam B is incident on the patterning device (e.g., mask)MA, which is held on the support structure (e.g., mask table) MT, and ispatterned by the patterning device. After being reflected from thepatterning device (e.g. mask) MA, the radiation beam B passes throughthe projection system PS, which focuses the beam onto a target portion Cof the substrate W. With the aid of the second positioner PW andposition sensor PS2 (e.g. an interferometric device, linear encoder orcapacitive sensor), the substrate table WT can be moved accurately, e.g.so as to position different target portions C in the path of theradiation beam B. Similarly, the first positioner PM and anotherposition sensor PS1 can be used to accurately position the patterningdevice (e.g. mask) MA with respect to the path of the radiation beam B.Patterning device (e.g. mask) MA and substrate W may be aligned usingpatterning device alignment marks M1, M2 and substrate alignment marksP1, P2.

The depicted apparatus 1000 could be used in at least one of thefollowing modes:

1. In step mode, the support structure (e.g. mask table) MT and thesubstrate table WT are kept essentially stationary, while an entirepattern imparted to the radiation beam is projected onto a targetportion C at one time (i.e. a single static exposure). The substratetable WT is then shifted in the X and/or Y direction so that a differenttarget portion C can be exposed.

2. In scan mode, the support structure (e.g. mask table) MT and thesubstrate table WT are scanned synchronously while a pattern imparted tothe radiation beam is projected onto a target portion C (i.e. a singledynamic exposure). The velocity and direction of the substrate table WTrelative to the support structure (e.g. mask table) MT may be determinedby the (de-)magnification and image reversal characteristics of theprojection system PS.

3. In another mode, the support structure (e.g. mask table) MT is keptessentially stationary holding a programmable patterning device, and thesubstrate table WT is moved or scanned while a pattern imparted to theradiation beam is projected onto a target portion C. In this mode,generally a pulsed radiation source is employed and the programmablepatterning device is updated as required after each movement of thesubstrate table WT or in between successive radiation pulses during ascan. This mode of operation can be readily applied to mask lesslithography that utilizes programmable patterning device, such as aprogrammable mirror array of a type as referred to above.

FIG. 19 shows the apparatus 1000 in more detail, including the sourcecollector module SO, the illumination system IL, and the projectionsystem PS. The source collector module SO is constructed and arrangedsuch that a vacuum environment can be maintained in an enclosingstructure 220 of the source collector module SO. A EUV radiationemitting plasma 210 may be formed by a discharge produced plasma source.EUV radiation may be produced by a gas or vapor, for example Xe gas, Livapor or Sn vapor in which the very hot plasma 210 is created to emitradiation in the EUV range of the electromagnetic spectrum. The very hotplasma 210 is created by, for example, an electrical discharge causingat least partially ionized plasma. Partial pressures of, for example, 10Pa of Xe, Li, Sn vapor or any other suitable gas or vapor may berequired for efficient generation of the radiation. In an embodiment, aplasma of excited tin (Sn) is provided to produce EUV radiation.

The radiation emitted by the hot plasma 210 is passed from a sourcechamber 211 into a collector chamber 212 via an optional gas barrier orcontaminant trap 230 (in some cases also referred to as contaminantbarrier or foil trap), which is positioned in or behind an opening insource chamber 211. The contaminant trap 230 may include a channelstructure. Contamination trap 230 may also include a gas barrier or acombination of a gas barrier and a channel structure. The contaminanttrap or contaminant barrier 230 further indicated herein at leastincludes a channel structure, as known in the art.

The collector chamber 211 may include a radiation collector CO which maybe a so-called grazing incidence collector. Radiation collector CO hasan upstream radiation collector side 251 and a downstream radiationcollector side 252. Radiation that traverses collector CO can bereflected off a grating spectral filter 240 to be focused in a virtualsource point IF along the optical axis indicated by the dot-dashed line‘O’. The virtual source point IF is commonly referred to as theintermediate focus, and the source collector module is arranged suchthat the intermediate focus IF is located at or near an opening 221 inthe enclosing structure 220. The virtual source point IF is an image ofthe radiation emitting plasma 210.

Subsequently the radiation traverses the illumination system IL, whichmay include a facetted field mirror device 22 and a facetted pupilmirror device 24 arranged to provide a desired angular distribution ofthe radiation beam 21, at the patterning device MA, as well as a desireduniformity of radiation intensity at the patterning device MA. Uponreflection of the beam of radiation 21 at the patterning device MA, heldby the support structure MT, a patterned beam 26 is formed and thepatterned beam 26 is imaged by the projection system PS via reflectiveelements 28, 30 onto a substrate W held by the substrate table WT.

More elements than shown may generally be present in illumination opticsunit IL and projection system PS. The grating spectral filter 240 mayoptionally be present, depending upon the type of lithographicapparatus. Further, there may be more mirrors present than those shownin the Figures, for example there may be 1-6 additional reflectiveelements present in the projection system PS than shown in FIG. 19.

Collector optic CO, as illustrated in FIG. 19, is depicted as a nestedcollector with grazing incidence reflectors 253, 254 and 255, just as anexample of a collector (or collector mirror). The grazing incidencereflectors 253, 254 and 255 are disposed axially symmetric around theoptical axis O and a collector optic CO of this type is preferably usedin combination with a discharge produced plasma source, often called aDPP source.

Alternatively, the source collector module SO may be part of an LPPradiation system as shown in FIG. 20. A laser LA is arranged to depositlaser energy into a fuel, such as xenon (Xe), tin (Sn) or lithium (Li),creating the highly ionized plasma 210 with electron temperatures ofseveral 10's of eV. The energetic radiation generated duringde-excitation and recombination of these ions is emitted from theplasma, collected by a near normal incidence collector optic CO andfocused onto the opening 221 in the enclosing structure 220.

The embodiments may further be described using the following clauses:

1. A method for determining a deformation of a resist in a patterningprocess, the method comprising:obtaining a resist deformation model of a resist having a pattern, theresist deformation model configured to simulate a fluid flow of theresist due to capillary forces acting on a contour of at least onefeature of the pattern; anddetermining, via a processor and the resist deformation model, adeformation of a resist pattern to be developed based on an inputpattern to the resist deformation model.2. The method of clause 1, wherein the resist deformation model is basedon a linearized Navier-Stokes flow equations.3. The method of clause 1 or clause 2, wherein the fluid flow ischaracterized by a Stokes flow, and/or a Hele-Shaw flow.4. The method of clause 1 or clause 2, wherein the fluid flow ischaracterized by a fundamental solution that decays with distance to aparticular location in the resist at least as fast as 1/R, and that iseither regular in the resist or only singular at the location in theresist.5. The method of any of clauses 1-4, determining the deformationcomprises:defining a plurality of vertices along the contour of the at least onefeature;determining a capillary force at a given vertex along the contour of theat least one feature; anddetermining a velocity flow field of the fluid flow due to the capillaryforce based on a superposition of a velocity response at one or more ofthe plurality of the vertices of the vertices along the contour of theat least one feature, and a boundary condition, wherein the velocityresponse is due to the capillary force acting at a vertex adjacent tothe given vertex.6. The method of any of clause 5, wherein the defining the plurality ofvertices comprises: distributing the plurality of vertices to make themevenly spaced while conserving an area or a volume of the contour of theat least one feature.7. The method of any of clauses 1-6, wherein the capillary force actingon the given vertex is a sum of tensions on either side of the givenvertex.8. The method of clause 5, wherein the velocity response ischaracterized by Stokeslet, wherein a Stokeslet is the velocity fielddue to a point force in a Stokes flow.9. The method of clause 5, wherein the determining the velocity flowfield further comprises:

decomposing the velocity field into a squeeze flow and a higher ordervelocity flow;

applying a boundary condition corresponding to the squeeze flow; and

removing the squeeze flow from the velocity field based on the boundarycondition.10. The method of clause 9, wherein the squeeze flow is a flow of resistdue to a net inward flux or a net outward flux through the resist domainboundaries, causing a large scale migration of the features.11. The method of any of clauses 7-10, wherein applying the boundarycondition comprises:setting a flow rate through the boundary of a resist domain to zero;and/orsetting a velocity across the boundary of the resist domain to no-fluxcondition.12. The method of clause 9, wherein the total inward flow rate throughthe resist domain boundaries is set to zero by providing a rotlet ofappropriate strength at one or more corners of a boundary of the resistdomain.13. The method of clause 10, wherein the rotlets of equal magnitude andalternating signs when traversing the domain boundary, at are placed atthe four corners of the rectangular resist domain.14. The method of any of clauses 5-13, determining a velocity flow fieldfurther comprising: obtaining velocity at the given vertex along thecontour of the at least one feature based on velocities of all othervertices caused due to the capillary force at each of the other verticesby multiplication with Stokeslets.15. The method of clause 5, wherein contributions to the velocity flowfield due to features far away from the given vertex are negligible.16. The method of any of clauses 5-13, further comprising:determining forces at a partial area within the resist; and

obtaining a deformation of the entire area of the resist based on thesimulation of the resist deformation model using forces at the partialarea.

17. The method of any of clauses 1-16, wherein the input pattern is agrayscale image and/or a binary image.18. The method of any of clauses 1-17, wherein the input pattern is adesign pattern, a resist image, a mask pattern, and/or an aerial image.19. The method of clause 18, further comprising generating the binaryimage, the generating comprising:

obtaining a patterning device pattern corresponding to the inputpattern;

producing, via simulation of a process model, an aerial image based onthe patterning device pattern; and

extracting boundaries of the pattern in the aerial image to generate thebinary image.

20. The method of any of clauses 1-19, further comprising:

computing, using the resist deformation model, a critical dimensionbetween a pair of locations disposed on a boundary of the at least onefeature in the developed resist pattern; and

calculating an error between the computed critical dimension and ameasured critical dimension of an actual developed resist pattern.

21. The method of clause 20, wherein the calculating the error comprisescomparing, using a cross-correlation matrix between a printed wafer dataand a simulated images.22. The method of any of clauses 1-21, wherein the deformation isdetermined at a plurality of locations, each location corresponding to apoint that lies on a boundary of a developed portion of the developedresist pattern for the input pattern.23. The method of any of clauses 1-22, wherein the resist is a negativetone resist or a positive tone resist, chemically or not-chemicallyamplified.24. A method for determining a parameter of a patterning process, themethod comprising:obtaining (i) a patterning process model that includes a resistdeformation model of a resist having a pattern, the resist deformationmodel configured to simulate a fluid flow of the resist due to capillaryforces acting on a contour of at least one feature of the pattern, and(ii) a target pattern;determining, via a processor, a resist pattern based on a simulation ofthe patterning process model with the target pattern as an input to thepatterning process model, wherein a difference exists between the resistpattern and the target pattern; anddetermining, via the processor, a value of a parameter of the patterningprocess based on the simulation of the patterning process, the value ofthe parameter being determined such that the difference between theresist pattern and the target pattern is reduced.25. The method of clause 24, wherein the parameter of the patterningprocess comprises at least one of dose, focus, and optical proximitycorrection.26. The method of any of clauses 24-25, further comprising:applying the value of the parameter of the patterning process to alithographic apparatus during the patterning process.27. A method for determining a deformation of a pattern to be formed ina patterning process, the method comprising:inputting, into a resist deformation model, pattern information relatingto the pattern to be formed, the model configured to simulatedeformation of a portion of a resist, the portion comprising a boundaryliquid layer located at a boundary between a developed region in theresist and a region of the resist surrounding the developed region,

wherein the model is configured to determine a first deformationcomponent of the boundary liquid layer caused by fluid flow of theboundary liquid layer and a second deformation component of the boundaryliquid layer caused by the fluid flow of the boundary liquid layer; and

determining, via a processor, the deformation of the pattern to beformed in the resist based on the input pattern information, wherein thedeformation comprises a combination of the first deformation componentand the second deformation component of the boundary liquid layer.28. The method of clause 27, wherein the boundary liquid layer has athickness smaller than a length of the developed region in the resist atthe boundary.29. The method of any of clauses 27-28, wherein the first deformationcomponent is determined in a horizontal plane based on a horizontalcomponent of a flow rate of the boundary liquid layer and the seconddeformation is determined in the horizontal plane based on a verticalcomponent of the flow rate of the boundary liquid layer.30. The method of any of clauses 27-29, wherein the model has definedtherein, a first boundary condition comprising a velocity gradientapproximately equal to zero at a free surface of the boundary liquidlayer, the free surface being opposite of the developed or open region.31. The method of any of clauses 27-30, wherein the model has definedtherein, a second boundary condition comprising a velocity approximatelyequal to zero at a surface at the developed or open region.32. The method of clause 31, wherein the model has defined therein, thevelocity as a function of thickness of the boundary liquid layer and apressure gradient across the thickness of the boundary liquid layer.33. The method of clause 32, wherein the model has defined therein, theflow rate of the boundary liquid layer as a function of an integrationof the velocity over the thickness of the boundary liquid layer.34. The method of clause 29, wherein the model has defined therein, thedeformation as a function of the flow rate of the boundary liquid layer.35. The method of any of clauses 27-34, wherein determining of thedeformation comprises:

determining the first deformation component in a first plane caused bythe horizontal component of the flow rate of the boundary liquid layer;and

determining a final deformation in the first plane by adjusting thefirst deformation component based on the vertical component of the flowrate of the boundary liquid layer in a second plane, the second plane isperpendicular to the first plane.

36. The method of clause 35, wherein the first plane is the horizontalplane.37. The method of any of clauses 27-36, wherein the determining thedeformation further comprises:defining a plurality of vertices along a contour of the developed oropen region in the resist;determining a capillary force at a given vertex along the contour; and

redistribution of the vertices such that (i) a volume of the developedregion in the resist is conserved, and (ii) a volume of the boundaryliquid layer conservation.

38. The method of clause 37, wherein the redistribution of the verticescomprises:

determining a time scale based on the horizontal component of the flowrate of the boundary liquid layer; and

moving, based on the time scale and the flow rate of the boundary layer,the plurality of vertices.

39. The method of clause 38, further comprising:

determining a rate of change of thickness of the boundary liquid layerbased on the flow rate of the boundary layer; and

adjusting, based on the redistributed vertices, another plurality ofvertices along the free surface based on rate of change of thickness ofthe boundary liquid layer.

40. The method of any of clauses 27-39, wherein the model is a thin-filmmodel simplified based on lubrication approximation, wherein thelubrication approximation comprises a vertical shear stress value ofzero.41. A method for determining a deformation of a pattern to be formed ina patterning process, the method comprising:inputting, into a resist deformation model, pattern information relatingto the pattern to be formed, the model configured to simulatedeformation of a portion of a resist, the portion comprising a boundaryliquid layer located at a boundary between a developed region in theresist and a region of the resist surrounding the developed region,wherein the model is configured to determine a deformation of theboundary liquid layer caused by a horizontal fluid flow of the boundaryliquid layer; anddetermining, via a processor, the deformation of the pattern to beformed in the resist by simulating the resist deformation model based onthe input pattern information.42. The method of clause 41, wherein the boundary liquid layer has athickness smaller than a length of the developed region in the resist atthe boundary.43. A non-transitory computer program product comprisingmachine-readable instructions for causing a processor to causeperformance of the method of any of clauses 1-42.

Although specific reference may be made in this text to the manufactureof devices such as ICs, it should be explicitly understood that thedescription herein has many other possible applications. For example, itmay be employed in the manufacture of integrated optical systems,guidance and detection patterns for magnetic domain memories,liquid-crystal display panels, thin-film magnetic heads, etc. Theskilled artisan will appreciate that, in the context of such alternativeapplications, any use of the terms “reticle”, “wafer” or “die” in thistext should be considered as interchangeable with the more general terms“mask”, “substrate” and “target portion”, respectively.

It is noted that the terms “mask”, “reticle”, and “patterning device”are utilized interchangeably herein. Also, a person skilled in the artwill recognize that, especially in the context of lithographysimulation/optimization, the term “mask”/“patterning device” and “designlayout” can be used interchangeably, as in lithographysimulation/optimization, a physical patterning device is not necessarilyused but a design layout can be used to represent a physical patterningdevice.

In the present document, the terms “radiation” and “beam” are used toencompass all types of electromagnetic radiation, including ultravioletradiation (e.g. with a wavelength of 365, 248, 193, 157 or 126 nm) andextreme ultra-violet radiation (EUV), e.g. having a wavelength in therange 5-20 nm).

The terms “optimizing” and “optimization” as used herein mean adjustinga lithographic projection apparatus and/or a patterning process suchthat results and/or processes of the patterning process (such aslithography) have a more desirable characteristic, such as higheraccuracy of projection of design layouts on a substrate, a largerprocess window, etc. The terms “optimizing” and “optimization” do notnecessarily require that results and/or processes of lithography havethe most desirable characteristics, such as highest accuracy ofprojection of design layouts on a substrate, largest process window,etc.

The patterning device referred to above comprises or can form designlayouts. The design layouts can be generated utilizing CAD(computer-aided design) programs, this process often being referred toas EDA (electronic design automation). Most CAD programs follow a set ofpredetermined design rules in order to create functional designlayouts/patterning devices. These rules are set by processing and designlimitations. For example, design rules define the space tolerancebetween circuit devices (such as gates, capacitors, etc.) orinterconnect lines, so as to ensure that the circuit devices or lines donot interact with one another in an undesirable way. The design rulelimitations are typically referred to as “critical dimensions” (CD). Acritical dimension of a circuit can be defined as the smallest width ofa line or hole or the smallest space between two lines or two holes.Thus, the CD determines the overall size and density of the designedcircuit. Of course, one of the goals in integrated circuit fabricationis to faithfully reproduce the original circuit design on the substrate(via the patterning device).

The term “mask” or “patterning device” as employed in this text may bebroadly interpreted as referring to a generic patterning device that canbe used to endow an incoming radiation beam with a patternedcross-section, corresponding to a pattern that is to be created in atarget portion of the substrate; the term “light valve” can also be usedin this context. Besides the classic mask (transmissive or reflective;binary, phase-shifting, hybrid, etc.), examples of other such patterningdevices include a programmable mirror array and/or a programmable LCDarray.

The concepts disclosed herein may simulate or mathematically model anypatterning process, and may be especially useful with imagingtechnologies capable of producing increasingly shorter wavelengths.Examples of such imaging technologies already in use include EUV(extreme ultra violet), DUV lithography that is capable of producing a193 nm wavelength with the use of an ArF laser and/or a 157 nmwavelength with the use of a fluorine laser. Moreover, EUV lithographyis capable of producing wavelengths within a range of about 5 nm toabout 20 nm by using, e.g., a synchrotron or by hitting a material(either solid or a plasma) with high energy electrons in order toproduce photons within this range.

While the concepts disclosed herein may be used for patterning processesinvolving imaging on a substrate such as a silicon wafer, it shall beunderstood that the disclosed concepts may be used with any type oflithographic systems, e.g., those used for imaging on substrates otherthan silicon wafers.

In block diagrams, illustrated components are depicted as discretefunctional blocks, but embodiments are not limited to systems in whichthe functionality described herein is organized as illustrated. Thefunctionality provided by each of the components may be provided bysoftware or hardware modules that are differently organized than ispresently depicted, for example such software or hardware may beintermingled, conjoined, replicated, broken up, distributed (e.g. withina data center or geographically), or otherwise differently organized.The functionality described herein may be provided by one or moreprocessors of one or more computers executing code stored on a tangible,non-transitory, machine readable medium. In some cases, third partycontent delivery networks may host some or all of the informationconveyed over networks, in which case, to the extent information (e.g.,content) is said to be supplied or otherwise provided, the informationmay be provided by sending instructions to retrieve that informationfrom a content delivery network.

Unless specifically stated otherwise, as apparent from the discussion,it is appreciated that throughout this specification discussionsutilizing terms such as “processing,” “computing,” “calculating,”“determining” or the like refer to actions or processes of a specificapparatus, such as a special purpose computer or a similar specialpurpose electronic processing/computing device.

The reader should appreciate that the present application describesseveral inventions. Rather than separating those inventions intomultiple isolated patent applications, these inventions have beengrouped into a single document because their related subject matterlends itself to economies in the application process. But the distinctadvantages and aspects of such inventions should not be conflated. Insome cases, embodiments address all of the deficiencies noted herein,but it should be understood that the inventions are independentlyuseful, and some embodiments address only a subset of such problems oroffer other, unmentioned benefits that will be apparent to those ofskill in the art reviewing the present disclosure. Due to costsconstraints, some inventions disclosed herein may not be presentlyclaimed and may be claimed in later filings, such as continuationapplications or by amending the present claims. Similarly, due to spaceconstraints, neither the Abstract nor the Summary sections of thepresent document should be taken as containing a comprehensive listingof all such inventions or all aspects of such inventions.

It should be understood that the description and the drawings are notintended to limit the present disclosure to the particular formdisclosed, but to the contrary, the intention is to cover allmodifications, equivalents, and alternatives falling within the spiritand scope of the inventions as defined by the appended claims.

Modifications and alternative embodiments of various aspects of theinventions will be apparent to those skilled in the art in view of thisdescription. Accordingly, this description and the drawings are to beconstrued as illustrative only and are for the purpose of teaching thoseskilled in the art the general manner of carrying out the inventions. Itis to be understood that the forms of the inventions shown and describedherein are to be taken as examples of embodiments. Elements andmaterials may be substituted for those illustrated and described herein,parts and processes may be reversed or omitted, certain features may beutilized independently, and embodiments or features of embodiments maybe combined, all as would be apparent to one skilled in the art afterhaving the benefit of this description. Changes may be made in theelements described herein without departing from the spirit and scope ofthe invention as described in the following claims. Headings used hereinare for organizational purposes only and are not meant to be used tolimit the scope of the description.

As used throughout this application, the word “may” is used in apermissive sense (i.e., meaning having the potential to), rather thanthe mandatory sense (i.e., meaning must). The words “include”,“including”, and “includes” and the like mean including, but not limitedto. As used throughout this application, the singular forms “a,” “an,”and “the” include plural referents unless the content explicitlyindicates otherwise. Thus, for example, reference to “an” element or “a”element includes a combination of two or more elements, notwithstandinguse of other terms and phrases for one or more elements, such as “one ormore.” The term “or” is, unless indicated otherwise, non-exclusive,i.e., encompassing both “and” and “or.” Terms describing conditionalrelationships, e.g., “in response to X, Y,” “upon X, Y,”, “if X, Y,”“when X, Y,” and the like, encompass causal relationships in which theantecedent is a necessary causal condition, the antecedent is asufficient causal condition, or the antecedent is a contributory causalcondition of the consequent, e.g., “state X occurs upon condition Yobtaining” is generic to “X occurs solely upon Y” and “X occurs upon Yand Z.” Such conditional relationships are not limited to consequencesthat instantly follow the antecedent obtaining, as some consequences maybe delayed, and in conditional statements, antecedents are connected totheir consequents, e.g., the antecedent is relevant to the likelihood ofthe consequent occurring. Statements in which a plurality of attributesor functions are mapped to a plurality of objects (e.g., one or moreprocessors performing steps A, B, C, and D) encompasses both all suchattributes or functions being mapped to all such objects and subsets ofthe attributes or functions being mapped to subsets of the attributes orfunctions (e.g., both all processors each performing steps A-D, and acase in which processor 1 performs step A, processor 2 performs step Band part of step C, and processor 3 performs part of step C and step D),unless otherwise indicated. Further, unless otherwise indicated,statements that one value or action is “based on” another condition orvalue encompass both instances in which the condition or value is thesole factor and instances in which the condition or value is one factoramong a plurality of factors. Unless otherwise indicated, statementsthat “each” instance of some collection have some property should not beread to exclude cases where some otherwise identical or similar membersof a larger collection do not have the property, i.e., each does notnecessarily mean each and every. References to selection from a rangeincludes the end points of the range.

In the above description, any processes, descriptions or blocks inflowcharts should be understood as representing modules, segments orportions of code which include one or more executable instructions forimplementing specific logical functions or steps in the process, andalternate implementations are included within the scope of the exemplaryembodiments of the present advancements in which functions can beexecuted out of order from that shown or discussed, includingsubstantially concurrently or in reverse order, depending upon thefunctionality involved, as would be understood by those skilled in theart.

While certain embodiments have been described, these embodiments havebeen presented by way of example only, and are not intended to limit thescope of the present disclosures. Indeed, the novel methods, apparatusesand systems described herein can be embodied in a variety of otherforms; furthermore, various omissions, substitutions and changes in theform of the methods, apparatuses and systems described herein can bemade without departing from the spirit of the present disclosures. Theaccompanying claims and their equivalents are intended to cover suchforms or modifications as would fall within the scope and spirit of thepresent disclosures.

1. A method comprising: obtaining a resist deformation model for aresist of a patterning process, the resist deformation model configuredto simulate a fluid flow of the resist due to capillary forces acting ona contour of at least one feature of a pattern of the resist; anddetermining, via a hardware processor and the resist deformation model,a deformation of a resist pattern to be developed based on an inputpattern to the resist deformation model.
 2. The method of claim 1,wherein the resist deformation model is based on Navier-Stokes flowequations, and wherein the fluid flow is characterized by a Stokes flowand/or a Hele-Shaw flow, and wherein the fluid flow is characterized bya fundamental solution that decays with distance to a particularlocation in the resist, and that is either regular in the resist or onlysingular at the location in the resist.
 3. The method of claim 1,wherein the determining the deformation comprises: defining a pluralityof vertices along the contour of the at least one feature; determining acapillary force at a given vertex of the plurality of vertices; anddetermining a velocity flow field of the fluid flow due to the capillaryforce based on a superposition of a velocity response at one or more ofthe plurality of the vertices, and based on a boundary condition,wherein the velocity response is due to a capillary force acting at avertex adjacent to the given vertex.
 4. The method of any of claim 3,wherein the defining the plurality of vertices comprises distributingthe plurality of vertices to make them evenly spaced while conserving anarea or a volume of the contour of the at least one feature, wherein thecapillary force acting on the given vertex is a sum of tensions oneither side of the given vertex.
 5. The method of claim 3, wherein thevelocity response is characterized by Stokeslet, wherein a Stokesletrepresents a velocity flow field due to a point force in a Stokes flow.6. The method of claim 3, wherein the determining the velocity flowfield further comprises: decomposing the velocity flow field into asqueeze flow and a higher order velocity flow; applying a boundarycondition corresponding to the squeeze flow; and removing the squeezeflow from the velocity flow field based on the boundary conditioncorresponding to the squeeze flow.
 7. The method of claim 6, wherein thesqueeze flow is a flow of resist due to a net inward flux or a netoutward flux through resist domain boundaries, causing a large scalemigration of features in the resist pattern.
 8. The method of claim 6,wherein applying the boundary condition corresponding to the squeezeflow comprises: setting a flow rate through a boundary of a resistdomain to zero; and/or setting a velocity across a boundary of theresist domain to a no-flux condition.
 9. The method of claim 7, whereina total inward flow rate through the resist domain boundaries is set tozero by providing a rotlet of selected strength at one or more cornersof a boundary of the resist domain.
 10. The method of claim 3, whereinthe determining a velocity flow field further comprises obtaining avelocity at the given vertex along the contour of the at least onefeature based on velocities of all other vertices caused due to thecapillary force at each of the other vertices by multiplication withStokeslets, and wherein contributions to the velocity flow field due tofeatures far away from the given vertex are neglected.
 11. The method ofclaim 3, further comprising: determining forces at a partial area withinthe resist; and obtaining a deformation of an entire area of the resistbased on a simulation of the resist deformation model using the forcesat the partial area.
 12. The method of claim 1, wherein the inputpattern is a design pattern, a resist image, a mask pattern, and/or anaerial image, wherein the resist is a negative tone resist or a positivetone resist, chemically or not-chemically amplified, and wherein furtherthe input pattern is a grayscale image and/or a binary image.
 13. Themethod of claim 1, further comprising: computing, using the resistdeformation model, a critical dimension between a pair of locationsdisposed on a boundary of the at least one feature in the developedresist pattern; and calculating an error between the computed criticaldimension and a measured critical dimension of an actual developedresist pattern, wherein the calculating the error comprises comparing,using a cross-correlation matrix, between printed wafer data and asimulated image.
 14. The method of claim 1, wherein the deformation isdetermined at a plurality of locations, each location corresponding to apoint that lies on a boundary of a developed portion of the developedresist pattern for the input pattern.
 15. A non-transitory computerprogram product comprising machine-readable instructions therein, theinstructions, when executed by processor system, configured to cause theprocessor system to at least: obtain a resist deformation model for aresist of a patterning process, the resist deformation model configuredto simulate a fluid flow of the resist due to capillary forces acting ona contour of at least one feature of a pattern of the resist; anddetermine, via the resist deformation model, a deformation of a resistpattern to be developed based on an input pattern to the resistdeformation model.
 16. The computer program product of claim 15, whereinthe resist deformation model is based on Navier-Stokes flow equations,and wherein the fluid flow is characterized by a Stokes flow and/or aHele-Shaw flow, and wherein the fluid flow is characterized by afundamental solution that decays with distance to a particular locationin the resist and that is either regular in the resist or only singularat the location in the resist.
 17. The computer program product of claim15, wherein the instructions configured to cause the processor system todetermine the deformation are further configured to cause the processorsystem to: define a plurality of vertices along the contour of the atleast one feature; determine a capillary force at a given vertex of theplurality of vertices; and determine a velocity flow field of the fluidflow due to the capillary force based on a superposition of a velocityresponse at one or more of the plurality of the vertices, and based on aboundary condition, wherein the velocity response is due to a capillaryforce acting at a vertex adjacent to the given vertex.
 18. The computerprogram product of claim 15, wherein the instructions configured tocause the processor system to define the plurality of vertices arefurther configured to cause the processor system to distribute theplurality of vertices to make them evenly spaced while conserving anarea or a volume of the contour of the at least one feature, wherein thecapillary force acting on the given vertex is a sum of tensions oneither side of the given vertex.
 19. The computer program product ofclaim 15, wherein the instructions are further configured to cause theprocessor system to: compute, using the resist deformation model, acritical dimension between a pair of locations disposed on a boundary ofthe at least one feature in the developed resist pattern; and calculatean error between the computed critical dimension and a measured criticaldimension of an actual developed resist pattern, wherein the calculatingthe error comprises comparing, using a cross-correlation matrix, betweenprinted wafer data and a simulated image.
 20. The computer programproduct of claim 15, wherein the deformation is determined at aplurality of locations, each location corresponding to a point that lieson a boundary of a developed portion of the developed resist pattern forthe input pattern.